Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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Answers to problems 455
4.9:
(l + 1)
dI
0
l+1
(τ )
dτ
+l
dI
0
l−1
(τ )
dτ
+
[
ω
0
p
l
− (2l + 1)
]
I
0
l
(τ )
=−
ω
0
2π
S
0
exp
−
τ
µ
0
p
0
l
P
0
l
(−µ
0
) − 2(1 − ω
o
)B(τ )δ
0l
Chapter 6
6.4:
E
+
(τ ) =
1
1 + ατ
c
[
E
+
(τ
c
)(1 + ατ) + E
−
(0)α(τ
c
− τ )
]
E
−
(τ ) =
1
1 + ατ
c
{E
+
(τ
c
)ατ + E
−
(0)
[
1 + α(τ
c
− τ )
]
}
with α
1
= α
2
= α.
6.5:
C
1
= A
E
+
(τ
c
) − γ
1
(B
0
+ B
1
τ
c
) − γ
2
B
1
−
exp(−λτ
c
)
β
2
(
E
−
(0) − γ
1
B
0
+ γ
2
B
1
)
(a) C
2
= A
exp(λτ
c
)
β
2
(E
−
(0) − γ
1
B
0
+ γ
2
B
1
) −
β
1
β
2
[
E
+
(τ
c
) − γ
1
(B
0
+ B
1
τ
c
) − γ
2
B
1
]
4
A =
exp(λτ
c
) −
β
1
β
2
exp(−λτ
c
)
−1
, β
1
=
α
2
α
1
+ λ
, β
2
=
α
2
α
1
− λ
(b)
d
dτ
(E
+
− E
−
) = C
1
(1 − β
1
)λ exp(λτ ) − C
2
(1 − β
2
)λ exp(−λτ )
6.6:
β
1 j
=
α
2 j
α
1 j
+ λ
, β
2 j
=
α
2 j
α
1 j
− λ
(i) τ = 0:
E
−
(0) = C
11
β
11
+ C
21
β
21
+ γ
11
B
01
− γ
21
B
11
(ii) τ = T
1
:
C
11
β
11
exp
(
λ
1
T
1
)
+ C
21
β
21
exp
(
−λ
1
T
1
)
+ γ
11
(
B
01
+ B
11
T
1
)
− γ
21
B
11
= C
12
β
12
+ C
22
β
22
+ γ
12
B
02
− γ
22
B
12
C
11
exp
(
λ
1
T
1
)
+ C
21
exp
(
−λ
1
T
1
)
+ γ
11
(
B
01
+ B
11
T
1
)
+ γ
21
B
11
= C
12
+ C
22
+ γ
12
B
02
+ γ
22
B
12
456 Answers to problems
(iii) τ = T
1
+ T
2
:
C
12
β
12
exp
(
λ
2
T
2
)
+ C
22
β
22
exp
(
−λ
2
T
2
)
+ γ
12
(
B
02
+ B
12
T
2
)
− γ
22
B
12
= C
13
β
13
+ C
23
β
23
+ γ
13
B
03
− γ
23
B
13
C
12
exp
(
λ
2
T
2
)
+ C
22
exp
(
−λ
2
T
2
)
+ γ
12
(
B
02
+ B
12
T
2
)
+ γ
22
B
12
= C
13
+ C
23
+ γ
13
B
03
+ γ
23
B
13
(iv) τ = T
1
+ T
2
+ T
3
:
E
+
(T
1
+ T
2
+ T
3
) = C
13
exp
(
λ
3
T
3
)
+ C
23
exp
(
−λ
3
T
3
)
+ γ
13
(
B
03
+ B
13
T
3
)
+ γ
23
B
13
Evaluating from these six equations the six unkown quantities C
ij
, i = 1, 2, j =
1, 2, 3 and substituting the result into (6.95) solves the problem.
6.7:
E
+
(τ ) = E
+
(τ
c
)2E
3
(τ
c
− τ ) + π B
0
[
1 − 2E
3
(τ
c
− τ )
]
E
−
(τ ) = E
−
(0)2E
3
(τ ) + π B
0
[
1 − 2E
3
(τ )
]
6.9:
T =−17
◦
C, effective radiation temperature. The difference of 31
◦
C can be
explained by the fact that in reality the atmosphere interacts with the radiation field.
6.10:
(a) T
4
s
=
2(1 − A
tot
) −
¯
A
σ (2 − ε)
S
0
4
, T
4
a
=
ε(1 − A
tot
−
¯
A) +
¯
A
εσ(2 − ε)
S
0
4
(b) T
s
= 273.9K,T
a
= 248.2K.
(c) T
s
= 302.0K,T
a
= 254.0K.
(d) ε = 0.87, T
a
= 249.8K.
Chapter 7
7.2:
(a) 5.7 hPa
(b) 20.3 hPa
7.3:
T = 25.04%
7.4:
A = 2π yL(
¯
u) + [1 − exp(−8
¯
uy
2
)] −
-
8π
¯
uy
2
erf(2y
√
2
¯
u) with
y =
α
L
ν
,
¯
u =
Su
2πα
L
and erf(x) =
2
√
π
x
0
exp(−t
2
)dt is the error function.
Answers to problems 457
7.6:
T = 34.17%
7.8:
S
ν
(p
0
) = S
ν,0
exp
−
2c
gas
3µ
0
g
p
0
k
ν,0
where c
gas
is the concentration of the absorbing
gas.
E
0,t
= 3600(1.0 − 0.31)2πµ
0
∞
0
S
ν
(p
0
)dν Jm
−2
.
7.11:
T = 81.81%
7.12:
f (k) =
S
πkδ
Sk
πα
L
− k
2
−1/2
7.13:
At 0 km:
∂T
rad
∂t
=−2.29 K day
−1
, at 1 km:
∂T
rad
∂t
=−1.62 K day
−1
,
at 3 km:
∂T
rad
∂t
=−0.57 K day
−1
.
7.14:
Lines do not overlap.
k
ν,i
=
/
k
max
1 −
|ν−ν
i
|
α
,ν
i
− α ≤ ν ≤ ν
i
+ α
0 elsewhere
, i = 1, 2
W
1
= W
2
= 2α
'
1 −
1
k
max
u
[1 − exp(−k
max
u)]
(
, W = W
1
+ W
2
= 2W
1
lim
u→∞
W
1
= 2α, lim
u→0
W
1
= 0, =⇒ lim
u→∞
W = 4α, lim
u→0
W = 0
7.15:
W =
Sc
g
(
p
1
− p
2
)
Chapter 10
10.1:
Left circularly polarized
458 Answers to problems
10.2:
E =
E
0
√
2
(−i + j) cos
k
√
2
(x + y) − ωt
10.4:
(a) Left-handed elliptically polarized
(b)
n
=2
n
=1
n
=0
n
=3
n
=6
n
=5
n
=4
n
=7
10.10:
F
11
=|A
2
|
2
, F
22
=|A
1
|
2
, F
33
=|A
1
||A
2
|cos δ
F
34
=−|A
1
||A
2
|sin δ, F
43
=|A
1
||A
2
|sin δ, F
44
=|A
1
||A
2
|cos δ
The remaining elements are zero.
Chapter 11
11.3:
z
m
= 0
List of frequently used symbols
In the following we will present a list of the most commonly used symbols of this
textbook. The units of all quantities are given in the so-called Syst
`
eme International
d’Unit
´
e (SI) or MKS system measuring lengths in meters, mass in kilograms and
time in seconds. The SI consists of the following seven fundamental units:
Physical quantity Unit name Unit abbreviation
Length meter m
Mass kilogram kg
Time second s
Current Ampere A
Temperature
1
Kelvin K
Luminous intensity candela cd
Amount of substance mole mol
1
Often temperature is also measured in
◦
C.
In addition to the seven basic units the following derived units are used:
Physical quantity Unit name Unit abbreviation
Solid angle steradian sr
Frequency Hertz Hz
Force Newton N
Pressure Pascal Pa
Energy Joule J
Power Watt W
Voltage Volt V
Magnetic flux density Tesla T
Charge Coulomb C
459
460 List of frequently used symbols
A : area (m
2
)
A(
¯
u) : absorption function
A
o
: average albedo of the cloud-free atmosphere
A
c
: average albedo of the overcast sky
A
g
: albedo of the ground
A
mn
: Mie coefficient
A
n
: Mie coefficient
A
u→l
: Einstein coefficient for spontaneous emission (s
−1
)
A
ν
: monochromatic absorption
A : vector potential (V m
−1
s)
A : coefficient matrix of two-stream method
A : transformation matrix for the Stokes vectors
A
m
: matrix used in spherical harmonics method
a
1
, a
2
: amplitudes of the electric vector (V m
−1
)
a
i
: coefficient to fit the transmission function
a
m
: expansion coefficient of the wave function
a
s
n
: expansion coefficient in the Mie theory
B
ν,g
: black body radiation of the ground (W m
−2
sr
−1
Hz
−1
)
B
mn
: Mie coefficient
B
n
: Mie coefficient
B
u→l
: Einstein coefficient for induced emission (m
3
J
−1
s
−2
)
B
l→u
: Einstein coefficient for absorption (m
3
J
−1
s
−2
)
B
λ
: black body radiance (W m
−3
sr
−1
)
B
ν
: black body radiance (W m
−2
sr
−1
Hz
−1
)
B : vector for polarized black body radiation (W m
−1
sr
−1
)
B : magnetic field vector (T)
B
±
: matrices for up- and downward radiation
B
m
: matrix used in spherical harmonics method
b : backscattered fraction of radiation
b
i
: coefficient to fit the transmission function
b
ij
: symbols used in Gaussian quadrature
b
s
n
: expansion coefficient in the Mie theory
C
abs
: absorption cross-section (m
2
)
C
ext
: extinction cross-section (m
2
)
C
f,l
: cloud forcing for long-wave radiation (W m
−2
)
C
f,s
: cloud forcing for short-wave radiation (W m
−2
)
C
i
: cloud fraction of layer i
c
p
: specific heat at constant pressure (m
2
s
−2
K
−1
)
C
sca
: scattering cross-section (m
2
)
c : speed of light in vacuum (m s
−1
)
D : dissociation energy (J)
D
p
: degree of polarization
D : displacement vector (A s m
−2
)
D
m
: matrix used in spherical harmonics method
E : radiative effect
E
0
: radiative effect of the base atmosphere
E
S
: radiative effect according to the Schwinger functional
E
−,ν
: downward directed radiative flux density (W m
−2
Hz
−1
)
E
+,ν
: upward directed radiative flux density (W m
−2
Hz
−1
)
List of frequently used symbols 461
E
net,ν
: net radiative flux density vector (W m
−2
Hz
−1
)
E
n
: exponential integral of order n
E
i,s
l,r
: components of the electric field vector (V m
−1
)
e : energy of a photon (J) or (eV)
e : specific internal energy of the air (J kg
−1
)
e : electric charge (C)
E : electric field vector (V m
−1
)
e
λ,ϕ,r
: unit vectors in λ, ϕ,r - direction
E : unit matrix
f : fraction of the forward scattered radiation
f (k) : probability density function for absorption
coefficient (m)
f (t) : time signal of a wave
f (ν − ν
0
) : profile of a spectral line
f
ν
: distribution function of photons (m
−3
sr
−1
Hz
−1
)
G(ω) : power spectrum
g : asymmetry parameter of the phase function
g(k) : cumulative probability function for absorption
coefficient k
g(ω) : Fourier transform of f (t)
g
l,u
: statistical weights of lower and upper states
G : matrix used in spherical harmonics method
H : Hamilton function
H(x) : Hopf function
H
n
(x) : Hermite polynomial of order n
H
1,2
n
: Hankel functions of the first and second kind and
of order n
H : quantum mechanical Hamilton operator (J)
H
: perturbation Hamilton operator (J)
I
ν
: radiance in a plane–parallel atmosphere (W m
−2
sr
−1
Hz
−1
)
I
m
ν
: m-th term in the development of I
ν
(W m
−2
sr
−1
Hz
−1
)
I
+
ν
: adjoint radiance
I
−,ν
: downward directed radiance (W m
−2
sr
−1
Hz
−1
)
I
+,ν
: upward directed radiance (W m
−2
sr
−1
Hz
−1
)
I
n
: modified Bessel function of first kind and of
order n
I
ν
: radiance (W m
−2
sr
−1
Hz
−1
)
I
m
±,ν
: up- and downward directed radiance vector (W m
−2
sr
−1
Hz
−1
)
i
1,2
: Mie intensity functions (sr
−1
)
J : rotational quantum number
J
ν
: source function for plane–parallel atmosphere (W m
−3
sr
−1
Hz
−1
)
J
e
ν
: source function for true emission (W m
−3
sr
−1
Hz
−1
)
J
m
ν
: m-th term in the development of J
ν
(W m
−3
sr
−1
Hz
−1
)
J
n
: Bessel function of first kind and of order n
j
ν
: emission coefficient (m
−3
s
−1
sr
−1
Hz
−1
)
J : angular momentum (kg m
2
s
−1
)
J : current density (A m
−2
)
J : source function vector (W m
−2
sr
−1
)
J
q
: heat flux (J m
−2
)
462 List of frequently used symbols
K : quantum number
K : kinetic energy (J)
k : Boltzmann’s constant (J K
−1
)
k : Hooke’s constant (kg s
−2
)
k : wave number (m
−1
)
k
0
: wave number in vacuum (m
−1
)
k
abs,ν
: absorption coefficient (m
−1
)
k
ext,ν
: extinction coefficient (m
−1
)
k
ν,D
: absorption coefficient of the Doppler line (m
−1
)
k
ν,L
: absorption coefficient of the Lorentz line (m
−1
)
k
ν,V
: absorption coefficient of the Voigt line (m
−1
)
k
sca,ν
: scattering coefficient (m
−1
)
k : wave number vector (m
−1
)
L : Laplace transform
L : Lagrange function
L : linear differential operator
L
+
: adjoint linear differential operator
L(
¯
u) : Ladenburg and Reiche function
M : magnetic quantum number
M, m : mass (kg)
m
e
o,m,n
: wave functions
M : magnetization (A m
−1
)
M : dipole moment (A s m)
M
ψ
: vector function to solve the vector wave equation
M : diagonal direction matrix
M
m
: matrix used in spherical harmonics method
N : complex index of refraction
N
l,u
: number of systems in lower and upper state (m
−3
)
N
ν
: total number of photons
n : number concentration of molecules (m
−3
)
n : normal unit vector
N
ψ
: vector function to solve the vector wave equation
N : Poynting vector (W m
−2
)
P(cos ) : phase function (sr
−1
)
˜
P(cos ) : scattering function (m
−1
sr
−1
)
P
HG
: Henyey–Greenstein phase function (sr
−1
)
P
l
: Legendre polynomial
P
m
l
: associated Legendre polynomial
¯
P(ϑ
) : probability distribution appearing in Monte Carlo
method
p : linear momentum (kg m s
−1
)
p, p
0
: air pressure (kg m
−1
s
−2
)
p(S) : distribution function for line intensities (m
2
)
p
c
: number of molecular collisions per unit time (s
−1
)
p
l
, p
m
l
: expansion coefficients of the phase function
p
∗
m
: δ-scaled expansion coefficient
P : polarization vector (A s m
−2
)
P : phase matrix
˜
P : scattering matrix
List of frequently used symbols 463
Q : source function (W m
−3
sr
−1
)
Q
g
: dimensionless surface source function
Q
+
: adjoint source function (W m
−3
sr
−1
)
Q
abs
: Mie efficiency factor for absorption
Q
ext
: Mie efficiency factor for extinction
Q
sca
: Mie efficiency factor for scattering
q : probability that a molecule does not collide per unit time
q
i
: generalized contravariant position coordinate (m)
q
k
: normal coordinate (m)
q
i
: covariant base vector
Q
m
±
: integral expressions in successive order scattering method (W m
−3
sr
−1
)
R(z) : response function
R
0
: gas constant for dry air (m
2
s
−2
K
−1
)
R
m
: expansion of phase function for m-th component of radiance
R
nm
: matrix element for transition n → m
r : particle radius (m)
r : position vector
r
c
: position vector for center of mass
r : reflectivity matrix
r
g
: ground reflection matrix
r
m
: discretized reflectivity matrix
S : scattering function
S : line intensity (m
−2
)
S
0
: direct sunlight (W m
−2
)
S
1,2
: Mie amplitude functions
S
J
: part of line intensity that depends only on rotational quantum
number J
S
ν
: direct solar radiative flux density (W m
−2
Hz
−1
)
T : transmission function
T
f
: flux transmission function
T : temperature (K)
t : tangential unit vector
t : transmissivity
t
m
: discretized transmission matrix
t : time (s)
U : Heavyside step function
u : absorber mass (kg m
−2
)
u : real scalar function
u
∗
: complex scalar function
ˆ
u
ν
: energy density (J m
−3
)
u
s,t
n
: scalar functions to solve Mie problem
V : potential energy (J)
V : volume (m
3
)
V (r) : Morse function (J)
v : specific volume (m
3
kg
−1
)
v : vibrational quantum number
v : velocity (m s
−1
)
v : velocity vector (m s
−1
)
W : equivalent width (m
−1
)
w : absorber mass (kg m
−2
)
464 List of frequently used symbols
w
j
: weights of the Gaussian quadrature formula
W : diagonal matrix of the Gaussian weights
X(µ) : Chandrasekhar’s X-function
X
nm
: x-component of matrix element of dipole moment for
transition n → m
(Asm)
x : size parameter
X
m
j
: matrix used in spherical harmonics method
Y (µ) : Chandrasekhar’s Y -function
Y
m
±
: terms used in successive order of scattering method
Z : partition function
Z
n
: cylinder function of order n
z
n
: spherical Bessel function of order n
α
D
: half-width of the Doppler line (Hz or m
−1
)
α
L
: half-width of the Lorentz line (Hz or m
−1
)
α
V
: half-width of the Voigt line (Hz or m
−1
)
α
i
: sensitivity coefficient for feedback (W m
−2
K
−1
)
δ : net sensitivity coefficient for feedback (W m
−2
)
δ : phase difference
δ(µ) : Dirac δ-function
δ
1,2
: phase angles to describe polarization
δ
ij
: Kronecker δ
δ
l,s
: sensitivity coefficients for long-wave and short-wave
radiation
(W m
−2
)
ε : emissivity
ε, ε
0
: permittivity, permittivity of free space (C V
−1
m
−1
)
ε
f
: flux emissivity function
ε
g
: emissivity of the ground
ε
r
: relative permittivity, dielectric constant
κ
abs,ν
: mass absorption coefficient (m
2
kg
−1
)
λ : wavelength (µm)
˜
λ : eigenvalues appearing in two-stream method
µ : reduced mass
µ, µ
0
: µ = cos ϑ, µ
0
= cos ϑ
0
µ, µ
0
: permeability, permeability of free space (V s A
−1
m
−1
)
µ
r
: relative permeability
ν : frequency (Hz)
˜ν : wave number (m
−1
)
: wave function
ρ : mass density of the dry air (kg m
−3
)
ρ : volume density of electric charge (A s m
−3
)
ρ
abs
: mass density of an absorber gas (kg m
−3
)
σ : electrical conductivity (A V
−1
m
−1
)
σ : mean line intensity (m
−2
)
σ : Stefan–Boltzmann constant (W m
−2
K
−4
)
σ
c
: collision cross-section (m
2
)
τ : optical depth
τ
∗
: δ-scaled optical depth
¯τ : average time between two molecular collisions (s)
: scattering angle