Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology

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7.2 Band models 225

interval at ν = 0isk

ν,i

. The contribution from all N lines in the spectral interval is



i=1

ν,i

. Therefore, the resultant transmission is given by

ν=0

(u) = exp



−



i=1

ν,i



i=1

exp(−k

ν,i

u) (7.84)

We have assumed that the lines in this superﬁcial spectrum are randomly spaced.

Due to the random spacing, the probability for a line to be located in the interval ν

to ν

+ dν

is dν

/δ. Consequently, the joint probability for lines to be located in

the intervals extending from ν

to ν

+ dν

, ν

to ν

+ dν

, ..., ν

to ν

+ dν

, ...,

to ν

+ dν

is given by

i=1

dν

/δ.

In order to allow for all possible arrangements of lines we must permit each

line to be located anywhere in the interval ν extending from −N δ/2toN δ/2.

Therefore, the average value of (7.84) over the interval is given by

T (u) =

i=1



N δ/2

−N δ/2

exp(−k

ν,i

u)dν

i=1



N δ/2

−N δ/2

dν

(7.85)

Assuming that each arrangement of lines is equally probable then (7.85) can be

simpliﬁed substantially. Instead of the product

i=1

we obtain the N -th power so

that (7.85) can be written as

T (u) =



N δ



N δ/2

−N δ/2

exp

(

−k

)

dν





1 −

N δ



N δ/2

−N δ/2

[1 − exp(−k

u)]dν



(7.86)

If we permit the number of lines to be very large (N →∞) then this equation

becomes the exponential function

T (u) = exp



−



∞

−∞

[1 − exp(−k

u)]dν



= exp



−

W (u)



(7.87)

where we have used the deﬁnition (7.50) of the equivalent width. It should be

noted that no particular line proﬁle has been speciﬁed in the derivation of (7.87).

In general, we will use the Lorentz proﬁle to specify the absorption coefﬁcient.

The artiﬁcial situation of lines with equal intensity will now be generalized to

a more realistic spectrum consisting of randomly distributed lines with different

intensities. Let us consider a certain frequency range containing N lines. Each of

these is assumed to belong to an inﬁnite array of randomly spaced lines of equal

226 Transmission in spectral lines and bands of lines

intensity and mean spacing N δ. According to (7.87) the mean transmission of array

i is given by

(u) = exp



−

(u)

N δ



(7.88)

Due to the assumption of randomness in each array we may expect that the multi-

plication property (7.49) also applies so that

T (u) =

i=1

(u) = exp



−

N δ



i=1

(u)



(7.89)

(u) = exp



−

W (u)



= exp[−A(u)]

(7.90)

The transmission function T

(u) is due to Goody (1952). Sometimes (7.90) is also

attributed to Mayer (1947) and one speaks of the Mayer–Goody model.

In the following we will show how to ﬁnd A(u). Since (7.90) refers to a realistic

spectrum, the line intensities of the N lines vary in strength. In order to account for

this situation various probability distributions p(S) for the distributions of S have

been proposed in the literature. We are going to discuss three prominent models.

Goody’s exponential model

Goody (1952) proposed the following distribution for the line intensities

p(S) =

exp

(

−S/σ

)

(7.91)

where σ is the average value of the line intensity in the given spectral interval. The

quantity p(S)dS then gives the percentage of lines in the spectral interval which

belong to the line intensity interval (S, S + dS). It is easily seen that the probability

distribution (7.91) fulﬁls the normalization condition



∞

p(S)dS = 1 (7.92)

The expectation value for the absorption A(u) due to the exponential model is

obtained from

A(u) =



∞

p(S)A(S, u)dS

with A(S, u) =



∞

−∞



1 − exp

(

− f

)



dν

(7.93)

7.2 Band models 227

Introducing (7.91) into (7.93) yields

A(u) =



∞

exp



−





∞

−∞

[1 − exp

(

− f

)

]dνdS =



∞

−∞

uσ

1 + f

uσ

dν

(7.94)

In the following we will use the line-shape factor of the Lorentz line (7.18) for

= 0. Employing the substitutions

u =

σ u

2πα

, y =

(7.95)

we obtain for the average absorption of the exponential model

A(u) =



∞

−∞

uα

+ α

(1 + 2

dν =

2π

√

1 + 2

(7.96)

The parameters σ and α

, or equivalently

u and y, have not been speciﬁed so

far. They can be determined by the so-called matching principle, i.e. one forces

the random band model to obey the limiting forms of the weak and the strong line

approximation (7.82) and (7.83) of the band model with non-overlapping lines.

Thus the matching principle can also be viewed as the closure assumption of the

random band models. For the weak line and strong line approximation we ﬁnd

using (7.96)

u  1: 2π y

u =

N δ



i=1

u  1: π y

√

u =

N δ



i=1

L,i

(7.97)

These two equations can be used to determine the unknown parameters

u and y.

The result is

u =

, y =

π N δ

with C =



i=1

, D =



i=1

L,i

(7.98)

The exponential distribution for S as assumed in the Goody model involves a

range of line intensities. It is clear that the strongest lines will be most important

for very short path lengths, while the very weak line intensities dominate the trans-

mission of radiation over very long path lengths. Therefore, in order to correctly

describe the mean transmission for arbitrary path lengths, the distribution function

228 Transmission in spectral lines and bands of lines

for S must be carefully chosen. While the exponential distribution leads to a simple

expression, in several cases the model provides rather inaccurate representations of

the line intensities in atmospheric bands. For example, in the case of water vapor,

the exponential distribution fails to account for the large number of weak lines in the

bands between 50 and 100

µm wavelength. For more details see Goody and Yung

(1989).

Godson’s inverse power model

Godson (1955) applied an inverse power law to describe the distribution of line

intensities as

p(S) =







for S

< S ≤ S

0 for S > S

(7.99)

The constant κ is a model parameter which is used to normalize the probability

distribution according to (7.92). The expectation value for absorption, employing

the inverse power law, is given by

A(u) = lim

→0



∞

−∞

[1 − exp

(

− f

)

]dνdS (7.100)

Since p(S) approaches inﬁnity if S

→ 0, the limiting value of the integral must be

considered. Using the substitutions

u =

2πα

, x =

κν

, y =

κα

(7.101)

we obtain for a Lorentz line the intermediate result

) = lim

→0



∞

−∞



1 − exp



−2

+ y



dxd

u (7.102)

where the upper bound

implicitly contains the dependence on u and S

.It

is convenient to carry out the integration over x ﬁrst. From Appendix 7.5.2 we

know



∞

−∞



1 − exp



−

+ y

)



dx = 2π y

u exp

(

−

)

(

u) + I

(

u)] (7.103)

Inserting this expression in (7.102) yields

) = 2π y lim

→0



exp

(

−

)

(

u) + I

(

u)]d

u (7.104)

7.2 Band models 229

The modiﬁed Bessel functions fulﬁll the following functional relations

(a)

−ν

(x)] = x

−ν

ν+1

(x)

(b)

[x exp

(

−x

)

(x) + I

(x))] = exp

(

−x

)

(x)

(7.105)

where ν is a ﬁxed real number. Integration by parts of the following integral gives



exp

(

−

)

(

u)d

u =



exp

(

−

)

(

u)d

u + exp

(

−

)

(



(7.106)

where (7.105a) with ν = 0 has been used. Substituting this equation into (7.104)

we obtain

) = 2π y lim

→0





2exp

(

−

)

(

u)d

u + 2π y exp

(

−

)

(





(7.107)

The integral on the right-hand side of this equation may be evaluated by means of

(7.105b) yielding

) = 2π y[2

exp

(

−

)

[

(

) + I

(

)

]

+ exp

(

−

)

(

)]

−2π y lim

→0

exp

(

−

)

[

(

) + I

(

)

]

+ exp

(

−

)

(

)] (7.108)

and hence

) = 2π y[2

exp

(

−

)

[

(

) + I

(

)

]

+ exp

(

−

)

(

)] − 2π y

(7.109)

Here, use was made of I

(0) = 1 and I

(0) = 0, see (7.71). This is the ﬁnal form

of the mean absorption of Godson’s model.

The unknown parameters y and

will again be determined by requiring in the

weak and strong line limit agreement between the Godson model and the band

model for nonoverlapping lines

 1: 2π y

N δ



i=1

 1: 4y

2π

N δ



i=1

(7.110)

230 Transmission in spectral lines and bands of lines

Solving these two equations gives

, y =

4N δ

(7.111)

where C and D are given by (7.98). The Godson model allows for a good

representation of weak lines in certain absorption bands and is usually more accurate

than the exponential law.

The Malkmus model

So far the most successful statistical model is due to Malkmus (1967). This model

is a combination of the Goody and the Godson model. Malkmus also observed

that the exponential distribution (7.91) substantially underestimates the number of

weak lines. If we consider only the Boltzmann factor in the line intensity formula

to be discussed in a later chapter, then S ∼ exp [−E /kT ], where E represents

the lower energy level in a molecular transition. From this relation he concludes

that dE/dS ∼ S

−1

. In many cases the number density of lines, n, is approximately

equally spaced with respect to variations in E, that is dn/dE ∼ const. The prob-

ability p(S)dS to ﬁnd lines with intensity S must be proportional to the change of

n versus S. Therefore, we obtain the relationship

p(S) ∼

∼

(7.112)

which shows that p(S) should vary as S

−1

. Indeed, it is this dominating inﬂu-

ence which determines the accuracy of the average band absorption. For this rea-

son Malkmus proposes a multiplicative combination of Goody’s and Godson’s

statistical models, that is

p(S) =

exp



−



(7.113)

where σ is the average value of the line intensities. For S = 0 the function p(S)of

the Godson and the Malkmus model is not deﬁned. Nevertheless, both models can

be applied by using a limiting procedure.

The average absorption A(u) due to the Malkmus model is now given by

A(u) = lim

ε→0



∞

exp



−





∞

−∞

[1 − exp

(

− f

)

]dνdS (7.114)

7.2 Band models 231

The integral over S will be evaluated by means of

lim

ε→0



∞

exp



−



[1 − exp

(

− f

)

]dS

= lim

ε→0



∞

exp



−



dS − lim



→0



∞



exp



−



=−lim

ε→0



ln ε −





−





±···



(7.115)

+ lim



→0



ln ε



−







−





±···

= ln(1 + σ f

where the substitutions S



= S(1 + σ f

u) and ε



= ε(1 + σ f

u) have been

introduced. Furthermore, it should be noted that the contributions of the upper

limit of the integrals over S and S



cancel so that only the difference of the lower

integral limits remains.

Substituting this result into (7.114) and applying the Lorentz line-shape factor

gives the average absorption of the Malkmus model

A(u) =



∞

−∞

ln(1 + σ f

u)dν = 2π y(

√

1 + 2

u − 1)

(7.116)

with y = α

/δ and

u = σ u/(2πα

). Finally, with the help of the matching proce-

dure we obtain

u  1: 2π y

u =

ν



i=1

u  1: 2π y

√

u =

ν



i=1

(7.117)

so that the unknown parameters y and

u are given as

u =

, y =

πν

(7.118)

The terms C and D are again given by (7.98). Applications of the Malkmus model

can be found, for example, in Crisp et al. (1986) and Lacis and Oinas (1991).

It should be observed that the absorption equations (7.96), (7.109) and (7.116)

have been derived on the assumption that only the intensities vary from line to

line. The model parameters

u and y employ the complete spectral data so that the

variation of the half-width from line to line is also taken into account.

232 Transmission in spectral lines and bands of lines

()

−110

Fig. 7.7 Line-shape factor of Elsasser’s regular band model with x = ν/δ.

The model parameters

u and y have been determined for the three statistical

models using different formulations of p(S). We are now in the position to employ

the Mayer–Goody transmission formula (7.90) with some conﬁdence. The spectral

data required to evaluate

u and y are given in Goody (1964a) or in Goody and Yung

(1989). This part of the sections on band models largely follow Goody’s (1964a)

excellent text.

7.2.4 Elsasser’s regular model

Spectra of linear molecules such as CO

often appear as a superposition of arrays

of more or less regularly spaced lines with nearly identical line shape. Due to

this observation Elsasser (1942) constructed a band model consisting of an inﬁnite

number of evenly spaced identical Lorentz lines. The absorption coefﬁcient is given

by the superposition of all lines, that is

ν,E

∞



n=−∞

(ν − nδ)

+ α

(7.119)

Figure 7.7 illustrates the line-shape factor f

(ν) = k

ν,E

/S of the Elsasser band

model. The spectral lines are separated by the distance δ.

The function k

ν,E

possesses an inﬁnite number of simple poles at ν = jδ ± iα

With the help of the Mittag–Lefﬂer theorem, see Appendix 7.6.3, the inﬁnite sum

can be stated as a closed expression involving periodic and hyperbolic functions.

This leads to the absorption coefﬁcient

k(s) =

sinh β

cosh β − cos s

(7.120)

7.2 Band models 233

where s = 2πν/δ and β = 2πα

/δ. Figure 7.7 suggests that k

varies periodically

within the bounds k

min

≤ k

max

. From (7.120) we immediately ﬁnd

min

sinh β

cosh β + 1

, k

max

sinh β

cosh β − 1

(7.121)

Next we will compute the average transmission of the regular band. Due to the

periodicity it is sufﬁcient to average over the interval [−δ/2,δ/2], i.e.

T (u) =



δ/2

−δ/2

exp

(

−k

)

dν =

2π



−π

exp

[

−k(s)u

]

(7.122)

It is convenient to determine the ﬁrst derivative of T with respect to u

=−

2π



−π

k(s)exp

[

−k(s)u

]

ds (7.123)

Substituting

cos φ =

1 − cosh β cos s

cosh β − cos s

(7.124)

gives

dφ =−

cosh β − cos φ

sinh β

ds =−k(φ)

ds (7.125)

The latter expression is valid since by means of simple trigonometric manipulations

it can be shown that

cosh β − cos φ

sinh β

cosh β − cos s

(7.126)

Using the above relations (7.123) may be rewritten as

2πδ



2π

exp



−

cosh β − cos φ

sinh β



dφ (7.127)

If we substitute in the argument of the exponential function the expression

y =

δ sinh β

(7.128)

we obtain

sinh β

2π

exp

(

−y cosh β

)



2π

exp

(

y cos φ

)

dφ (7.129)

234 Transmission in spectral lines and bands of lines

The Bessel functions J

of the ﬁrst kind can be expressed by the following

deﬁnite integral (Watson, 1980)

(x) =

(−i)

2π



−π

cos(nw)exp

(

ix cos w

)

dw, n ∈ N

(7.130)

Substitution of φ = π − w yields

(x) =

(−i)

2π



2π

cos(nπ − nφ)exp

[

ix cos(π − φ)

]

dφ (7.131)

which for n = 0 and y =−ix turns into

(iy) =

2π



2π

exp

(

y cos φ

)

dφ (7.132)

Hence (7.129) can be expressed as

= sinh β exp

(

−y cosh β

)

(iy) (7.133)

A ﬁnal substitution of Y = y sinh β = Su/δ then leads to a formula for the average

transmission of the Elsasser model

T (Y ) =



∞

exp(−Y



coth β)J





sinh β



(7.134)

There are no analytical solutions to this integral. However, results can be found by

means of numerical integration.

From Figure 7.7 we conclude that there exists an average value

k for the

absorption coefﬁcient k

. For the rapidly varying absorption coefﬁcients of real

spectra it is usually impossible to give a reliable value of

k. Formally we have

k =



δ/2

−δ/2

sinh β

cosh β − cos(2πν/δ)

dν (7.135)

Using the indeﬁnite integral



b + c cos ax

√

− c

tan

−1



(b − c) tan(ax /2)

√

− c



for b

> c

(7.136)

listed in integration tables, we ﬁnd the expected result

k =

(7.137)

This relation could have been guessed from Figure 7.7, since within the periodic

pattern of Lorentz lines the area under each line is S. Despite the fact that within