Absorption coe cient Friis equation revisited Absorption ... · PDF fileRadiative transfer in the atmosphere Transmission and Absorption Friis equation revisited Absorption coe cient

Radiative transferin the atmosphere

Transmission andAbsorption

Friis equation revisited

Absorption coefficient

Absorption byatmospheric gases

Transitions

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Related expressions

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Radiative transfer in the atmosphere

December 11, 2007






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Outline

Transmission and AbsorptionFriis equation revisitedAbsorption coefficient

Absorption by atmospheric gasesTransitionsLine shapeRelated expressions






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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?






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Transmission measurements at 94 GHz






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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






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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′.






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Index of refraction of water and ice in the infrared

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0.1 0.15 0.2 0.3 0.5 0.7 1 1.5 2 3 4 5 7 10 15 20 30 50

n rWavelength [µm]

(a) Index of Refraction of Water and Ice (Real Part)

Water (10o C)

Ice (-5o C)

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0.1 0.15 0.2 0.3 0.5 0.7 1 1.5 2 3 4 5 7 10 15 20 30 50

n i

Wavelength [µm]

(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






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Index of refraction of water

From Jackson: Classical electrodynamics






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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






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Rotational transitions

E0

E1

E2

Ener

gy

Ab

s. C

ross

-sec

tio

n

νν02ν01ν12

∆E12

∆E01

∆E02a)

b)


Position of lines: determined by ∆E of allowed transitionsRelative strength: determined by amount of molecules instates and likelihood that transition happens






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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







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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

http://spec.jpl.nasa.gov/






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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

Diatomic (N2, O2, CO)

Linear triatomic (CO2, N2O)

Symmetric stretch Bending Asymmetric stretchν1 ν3ν2

Nonlinear Triatomic (H2O, O3)

Symmetric stretch Bending Asymmetric stretchν1 ν3ν2

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






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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)

1. Natural broadening: Heisenberg uncertainty principle.Broadening negligible

2. Doppler broadening: Motions of molecules lead toDoppler-effect

3. Pressure broadening: Collisions between moleculesdisrupt natural transitions

Absorption coefficient: ka = nσν = nSf (ν − ν0)






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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






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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

]






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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


Linewidth of prominent lines in the atmosphere






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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)






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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






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Model by H.Liebe

N = (n − 1)106 ppm






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Model by H.Liebe

Attenuation due to rain

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Absorption coe cient Friis equation revisited Absorption ... · PDF fileRadiative transfer in the atmosphere Transmission and Absorption Friis equation revisited Absorption coe cient