Radiative transfer in the atmosphere Transmission and Absorption Friis equation revisited Absorption coefficient Absorption by atmospheric gases Transitions Line shape Related expressions 1 Radiative transfer in the atmosphere December 11, 2007 Radiative transfer in the atmosphere Transmission and Absorption Friis equation revisited Absorption coefficient Absorption by atmospheric gases Transitions Line shape Related expressions 2 Outline Transmission and Absorption Friis equation revisited Absorption coefficient Absorption by atmospheric gases Transitions Line shape Related expressions
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Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
1
Radiative transfer in the atmosphere
December 11, 2007
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
2
Outline
Transmission and AbsorptionFriis equation revisitedAbsorption coefficient
Absorption by atmospheric gasesTransitionsLine shapeRelated expressions
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
3
Transmission measurements
Communication between aligned transmitting and receivingantenna at a distance r
PR = PTGT
(4πr
λ
)−2
GR = PTGT1
LfGR = PTGTGf GR
Additional loss due to atmospheric attenuation:e−2αz = e−kaz
What is α resp. the absorption coefficient ka?
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
4
Transmission measurements at 94 GHz
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
5
Summary of relation of α with N and ε
dielectric constant ε = ε′ − iε′′
refractive index N = nr − ini
ε =ε
ε0= N2.
ε′ = n2r − n2
i ε′′ = 2nrni
nr =
√√(ε′)2 + (ε′′)2 + ε′
2
ni =
√√(ε′)2 + (ε′′)2 − ε′
2≈ ε′′
2√ε′
Absorption coefficient ka =4πni
λ
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
6
Kramers-Kronig relation
I Disperison and absorption are intimately linked
I A dispersive material must also be absorptive
I The relation is given by the Kramers-Kronig relation
I If one componenet (real or imaginery) is known at allfrequencies the other can be calculated
ε′(ω) = 1 +2
π
∞∫0
ω′ε′′(ω′)
ω′2 − ω2dω′
ε′′(ω) =2ω
π
∞∫0
1− ε′(ω′)ω′2 − ω2
dω′.
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
7
Index of refraction of water and ice in the infrared
(b) Index of Refraction of Water and Ice (Imag. Part)
Water (10o C)
Ice (-5o C)
1 µm
10 µm
0.1 mm
1 mm
1 cm
10 cm
1 m
10 m
100 m
0.2 0.3 0.5 0.7 1 1.5 2 3 4 5 7 10 15 20 30 50
Dep
th [m
]
Wavelength [µm]
Radiation Penetration Depth in Water and Ice
Water (10o C)
Ice (-5o C)
From G.Petty: A first course in atmospheric radiation
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
8
Index of refraction of water
From Jackson: Classical electrodynamics
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
9
Absorption by atmospheric gases
Basis of interaction processes between photons andmolecules → changes in internal energy:
I Changes in translational energy (temperature)
I Changes in rotational energy of polyatomic molecules
I Changes in vibrational energy of polyatomic molecules
I Changes in the distribution of molecular electric charges
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
10
Rotational transitions
E0
E1
E2
Ener
gy
Ab
s. C
ross
-sec
tio
n
νν02ν01ν12
∆E12
∆E01
∆E02a)
b)
From G.Petty: A first course in atmospheric radiation
Position of lines: determined by ∆E of allowed transitionsRelative strength: determined by amount of molecules instates and likelihood that transition happens
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
11
Rotational transitionsIn order for a molecule to interact with an electromagneticwave via rotational transitions, it must possess either amagnetic or electric dipole moment.E-field must have the capacity to exert a torque on themolecule
O OOxygen
N NNitrogen
O CCarbon Monoxide
OO CCarbon Dioxide
NNO
Nitrous Oxide
O
HH
Water
O
O O
Ozone
H
CH
HH
Methane
No(magneticdipole)
Yes
No
No
Yes
Yes
Yes
Yes
Molecule StructurePermanent Electirc
Dipole Moment?
linear
linear
linear
linear
linear
asymmetric top
asymmetric top
spherical top
From G.Petty: A first course in atmospheric radiation
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
12
Rotational spectra
Quantum mechanics gives us the energy levels for rigidrotator
EJ =J(J + 1)h2
8π2I
Selection rules define energy level differences
∆E = EJ+1 − EJ =h2
4π2I(J + 1)
and the corresponding frequencies
ν =∆E
h=
h
4π2I(J + 1) = 2B(J + 1)
with the rotational constant B = h8π2I
→ series of equally spaced lines separated by ∆ν = 2BThere exist spectroscopic data bases → JPL-catalog
Vibrational transitionsMolecular bonds are not rigid but behave like springs→ vibrationsFor polyatomic molecules a variety of vibrational modes exist→ superposition of set of normal modes
From G.Petty: A first course in atmosphericradiation
The ν2 and ν3 modes of CO2 at 15.0µm resp. 4.26µm areimportant for greenhouse warmingIn general we have to deal with vibration-rotation spectraIn the microwave region however mainly rotational transitions
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
14
Line shapesThree pieces of information are needed to characterize a linetransition:
1. Line position: Where does the line fall in thespectrum?
2. Line strength, S: How much total absorption?
3. Line shape: How is the absorption distributed aboutthe center of the line?
There are three processes responsible for line broadening,depending on local environmental conditions→ line shape function f (ν − ν0)
2. Doppler broadening: Motions of molecules lead toDoppler-effect
3. Pressure broadening: Collisions between moleculesdisrupt natural transitions
Absorption coefficient: ka = nσν = nSf (ν − ν0)
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
15
Doppler broadeningGas molecules are in constant motionDistribution of speed according Maxwell-BoltzmannLine shape for Doppler broadening
fD(ν − ν0) =1
αD√π
exp
(−(ν − ν0)2
α2D
)where
αD = ν0
√2kBT
mc2
Halfwidth at half max is α1/2
α1/2 = αD
√ln 2
I Doppler line shape is Gaussian
I Line width increases with temperature and decreaseswith molecular mass
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
16
Pressure broadeningBetween molecules there exist collisions affecting the lifetimeof states→ pressure broadening. No exact theory yet!Pressure broadening usually described by Lorentz line shape
fL(ν − ν0) =αL/π
(ν − ν0)2 + α2L
where
αL = α0
(p
p0
)(T0
T
)n
α0 at reference pressure p0 and temperature T0 from lab.Lorentz line shape has two deficiencies:
1. far lines poorly represented
2. only valid fro αL � α0
Better is the van Vleck-Weisskopf line shape
fVW =1
π
(ν
ν0
)2 [ αL
ν − ν0)2 + α2L
+αL
ν + ν0)2 + α2L
]
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
17
Comparing Doppler and pressure broadeningI Both mechanisms occur at all levels in the atmosphereI For typical values relative importance given by ratio ofαD to αL
αD
αL≈
[p0
α0c
√2kB
T0
]Tν0
p√
m∼ [5× 10−13mb Hz−1]
(ν0
p
)
0
20
40
60
80
100
104 1010109108107106105
α1/2 (Hz)
Z (k
m)
Pressure Broadening
Do
pp
ler B
road
enin
g
O2 λ=2.5 mm
CO2λ=15 µm λ=4.3 µm
From G.Petty: A first course in atmospheric radiation
Linewidth of prominent lines in the atmosphere
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
18
Opacity and transmittanceAtmosphere consists of many different species→ every constitutent contributes in its own way
ka =∑
i
ka,i =∑
i
niσa,i
Absorption along a path element ds is given by thelaw of Beer Lambert
dI (λ) = −ka(λ)I (λ)ds
which after integration yields
I (λ) = I0(λ)e−τa(λ,s)
Opacity τ : τa(λ, s) =∫s σa(λ)n(s)ds =
∫s ka(λ, s)ds
Penetration depth: δ = 1/ka
Transmittance: T (λ, s) = e−τa(λ,s)
Absorptivity: A(λ, s) = 1− T (λ, s)
Radiative transferin the atmosphere
Transmission andAbsorption
Friis equation revisited
Absorption coefficient
Absorption byatmospheric gases
Transitions
Line shape
Related expressions
19
Model by H.Liebe
Let’s go back to the transmission problemWhat is the attenuation in a horizontal propagation path inthe atmosphere?There exist models for atmospheric propagation by H.Liebeand Ph. Rosenkranz