Lesson16.ppt
Download
Report
Transcript Lesson16.ppt
METO 621
Lesson 16
Transmittance
• For monochromatic radiation the transmittance, T, is
given simply by
T ( ; ) e /
and
dT ( ; )
1 /
e
d
• The solution for the radiative transfer equation was:
Transmittance
I ( , , ) I ( *, , )e
( * ) /
*
1
S e ( ' ) / d '
Which is equivalent to
*
I ( , , ) I ( *, , )T ( * , ; )
dT ( ' , ; )
S d '
d '
similarly
dT ( , ' ; )
S d '
d '
0
I ( , , ) I (0, , )T ( ; )
Transmittance
• If we switch from the optical depth coordinate to
altitude, z, then we get
dT ( z, z ' ; )
I ( z, , ) I (0, , )T (0, z; )
S dz '
dz '
0
z
dT ( z , z ' ; )
I ( z , , ) I ( zT , , )T ( zT , z; )
S dz '
dz '
z
zT
• The change of sign before the integral is because
the integral range is reversed when moving from to z
Transmission in spectrally complex media
• So far we have defined the monochromatic transmittance,
T(). We can also define a beam absorptance ab() , where
ab() =1- T().
• But in radiative transfer we must consider transmission over
a broad spectral range, D. (Often called a band)
• In this case the mean band absorptance is defined as
1
a b
D
(1 e
k u
)d
D
• 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
TD
1
1 a D
D
k u
e d
D
Transmission in spectrally complex media
• If in the previous equation we replace the frequency
with wavenumber then the product
W a b D is called the equivalent width
• 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;
TD
1
D
S
a
u
L
d
exp
2
D (( 0 ) a L )
Transmission in spectrally complex media
• Let D be large enough to include all of the line,
then we can extend the limits of the integration to
±∞.
• Landenburg and Reichle showed that
1 x
TD 1
e J 0 (ix ) iJ 1 (ix )
D
Su
where x
J 0 and J1 are Bessel functions
2aL
Transmission in spectrally complex media
• For small x , known as the optically thin condition,
Su
T 1
D
for large x, optically thick,
2Su
2
T 1
1
a L Su
D
D 2x
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
T exp ycoth( )J 0 (iy sinh( )) dy
2aL
Su
where
and y
d
d sinh( )
optically thin
Su
T 1d
optically thick
2
T 1a L Su
d
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 D contains n lines
of mean separation d, i.e. D=nd
• Let the probability that line i has a line strength Si
be P(Si) where
P(S )dS 1
i
0
• 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 SP(S )dS
S
P(S )dS
0
• The transmission T over an interval D is given
by
1
Ti
D
d P(S )e
i
D
0
ki u
dSi
Statistical Band Model (Goody)
• For n lines the total transmission T is given by
T=T1T2T3T4…………..Tn
1
T
(D ) n
d d ................. d
1
D
1
D
n
D
0
0
0
x P( S1 )e k1u dS1 P( S 2 )e k 2u dS 2 ........ P( S n )e k nu dS1
1
ku
T
d P( S )e dS
n
(D ) D 0
1
1
D
d P(S )(1 e
D
0
ku
n
)dS
n
Statistical Band Model (Goody)
n
x
The equation has the form 1 which in
n
the limit equals e x
1
ku
T exp d P( S )(1 e )dS
d D o
assuming a Lorentz line shape, one gets
1 / 2
Su
Su
1
T exp
d aL
where S is the mean line strength.
Statistical Band Model (Goody)
• We have reduced the parameters needed to calculate T
to two
S
d
and
aL
d
• These two parameters are either derived by
fitting the values of T obtained from a line-by-line
calculation, or from experimental data.
weak line
Su
T exp
d
strong line
a
S
u
L
T exp
d
Summary of band models
K distribution technique
K distribution technique