METO 621 Lesson 16. Transmittance For monochromatic radiation the transmittance, T, is given simply...

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METO 621 Lesson 16
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Transcript of METO 621 Lesson 16. Transmittance For monochromatic radiation the transmittance, T, is given simply...

METO 621

Lesson 16

Transmittance

• For monochromatic radiation the transmittance, T, is given simply by

μτ

μτ

μτμτ

μτ

/

/

1);(

and

);(

−=

=

ed

dT

eT

• The solution for the radiative transfer equation was:

Transmittance

''

);,'();,(),*,(),,(

** τ

τμττμττφμτφμτ

τ

τνν dS

ddT

TII +++ ∫−=

'1

),*,(),,( /)'(*

/)*( τμ

φμτφμτ μτττ

τ

μττνν deSeII −−+−−++ ∫+=

Which is equivalent to

similarly

''

);',();(),,0(),,(

0

ττ

μττμτφμφμττ

νν dSd

dTTII −−− ∫+=

Transmittance

• If we switch from the optical depth coordinate to altitude, z, then we get

''

);',();,0(),,0(),,(

0

dzSdz

zzdTzTIzI

z+++ ∫+=

μμφμφμ νν

''

);',();,(),,(),,( dzS

dz

zzdTzzTzIzI

Tz

z

TT−−− ∫−=

μμφμφμ νν

• The change of sign before the integral is because the integral range is reversed when moving from τ to z

Transmission in spectrally complex media

• So far we have defined the monochromatic transmittance, Tν We can also define a beam absorptance b(ν) , where b(ν) =1- Tν

• But in radiative transfer we must consider transmission over a broad spectral range, ν(Often called a band)

• In this case the mean band absorptance is defined as

νν

ν

ν

deuk

b )1(1 −

Δ∫ −

Δ=⟩⟨

• Here u is the total mass along the path, and k is the monochromatic mass absorption coefficient.

Schematic for A

Transmission in spectrally complex media

• Similarly for the transmittance

⎟⎟⎠

⎞⎜⎜⎝

=⟩⟨−= ∫

ννν ν

ν ν deT uk1

1

Transmission in spectrally complex media

• If in the previous equation we replace the frequency with wavenumber then the product

widthequivalent thecalled is ν = bW

• The relationship of W to u is known as the curve of growth

Absorption in a Lorentz line shape

Transmission in spectrally complex media

• Consider a single line with a Lorentz line shape.

• As u gets larger the absorptance gets larger. However as u increases the center of the line absorbs all of the radiation, while the wings absorb only some of the radiation.

• At this point only the wings absorb.

• We can write the band transmittance as;

νννπ

ν ν

ν duS

TL

L∫Δ

Δ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

+−−

Δ=

))((exp

12

0

Transmission in spectrally complex media

• Let νbe large enough to include all of the line, then we can extend the limits of the integration to ±∞.

• Landenburg and Reichle showed that

[ ]

functions Bessel are and 2

where

)()(1

1

10

10

JJSu

x

ixiJixJeT

L

x

πα

νν

=

−Δ

−= −Δ

Transmission in spectrally complex media

• For small x , known as the optically thin condition,

Sux

SuT

SuT

Lνπν

ν

−=

−=

−=

21

22

1

,thickoptically ,xlargefor

1

Elsasser band model

Elsasser Band Model

• Elsasser and Culbertson (1960) assumed evenly spaced lines (d apart) of equal S that overlapped. They showed that the transmittance could be expressed as

[ ]

)sinh( and

2 where

))sinh(()coth(exp 0

β

παβ

ββ

d

Suy

d

dyiyJyT

L ==

−= ∫∞

∞−

SuTT Ld2

-1dSu

-1

thickopticallythinoptically

==

Statistical Band Model (Goody)

• Goody studied the water-vapor bands and noted the apparent random line positions and band strengths.

• Let us assume that the interval νcontains n lines of mean separation d, i.e. ν=nd

• Let the probability that line i has a line strength Si be P(Si) where

1)(0

=∫∞

dSSP i

• P is normally assumed to have a Poisson distribution

Statistical Band Model (Goody)

• For any line the most probable value of S is given by

∫∞

=

0

0

)(

)(

dSSP

dSSSP

S

• The transmission T over an interval νis given by

iuk

ii dSeSPdT i−∞

∫∫

= )(1

νν

Statistical Band Model (Goody)• For n lines the total transmission T is given by

T=T1T2T3T4…………..Tn

1

0

2

0

21

0

1

11

)(........)()(x

.................)(

1

21 dSeSPdSeSPdSeSP

dddT

ukn

ukuk

nn

n−∞

−∞

−∞

∫∫∫

∫∫∫=

ννν

νννν

n

ku

n

kun

dSeSPd

dSeSPdT

⎥⎦

⎤⎢⎣

⎡−

−=

⎥⎦

⎤⎢⎣

=

∫ ∫

∫ ∫

∞−

∞−

ν

ν

νν

νν

0

0

)1)((1

1

)()(

1

Statistical Band Model (Goody)

strength. linemean theis where

1exp

gets one shape, line Lorentz a assuming

)1)((1

exp

equalslimit the

in which 1 form thehasequation The

2/1

S

uS

d

uST

dSeSPdd

T

e

n

x

L

o

ku

x

n

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

⎥⎦

⎤⎢⎣

⎡−−≅

⎥⎦

⎤⎢⎣

⎡ −

∫ ∫

π

νν

Statistical Band Model (Goody)• We have reduced the parameters needed to calculate

T to two

dd

S Lπand

• These two parameters are either derived by fitting the values of T obtained from a line-by-line calculation, or from experimental data.

⎟⎟

⎜⎜

⎛−=⎟⎟

⎞⎜⎜⎝

⎛−=

d

uST

d

uST Lπα

exp exp

line strong line weak

Summary of band models

K distribution technique

K distribution technique