meto 621 lesson 16. transmittance for monochromatic radiation the transmittance, t, is given simply...
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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.
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
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 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
0ν
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