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

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7.4 Transmission in inhomogeneous atmospheres 255

where g is the cumulative probability density for the absorbing gas. For simplicity,

we have omitted in τ

(g) the pressure and temperature dependence. The contribu-

tion to the total optical depth in layer z resulting from Mie scattering is

τ

= τ

sca,M

+ τ

abs,M

(7.215)

The single scattering albedo in the homogeneous layer is deﬁned by

(g) =

τ

+ τ

sca,M

τ

+ τ

(g)

(7.216)

Let the Legendre expansion coefﬁcients for the Rayleigh and Mie scattering be

given by p

l,R

and p

l,M

. The Legendre expansion coefﬁcients for the phase function

for the combined action of air molecules and particle scattering is deﬁned by a

linear weighting of the expansion coefﬁcients of the individual components. The

weighting factor is the scattering optical depth of the respective material in layer

z, i.e.

τ

sca,M

l,M

+ τ

l,R

τ

sca,M

+ τ

(7.217)

For many cases the Mie phase function can be approximated by the Henyey–

Greenstein phase function, which for particle scattering depends on the asymmetry

factor only, see (6.41). Thus we obtain for the phase function the two relations

P(cos ) =



l=0

(cos ) =











(1 + cos

) Rayleigh scattering



l=0

(2l + 1)g

(cos ) Henyey–Greenstein

(7.218)

In these examples we have seen that the k-distribution method can be applied to

various simple models. It is difﬁcult to conceive any situation where this method

would fail.

7.4 Transmission in inhomogeneous atmospheres

In the previous sections we have derived various transmission functions for model

spectra of absorbing atmospheres. These transmission functions applying to homo-

geneous gaseous layers can be extended to simulate the transmission through inho-

mogeneous atmospheres. Ordinarily this involves coupled integrations over the

atmospheric path and the wave number. In order to avoid such extremely laborious

integrations to handle the transfer problem, it is customary to adopt approximate

A detailed description of the Mie theory will be given in a later chapter.

256 Transmission in spectral lines and bands of lines

techniques by means of scaling procedures which attempt to decouple these two

types of integrations. In earlier years radiation charts and the emissivity method

were used to study the radiative properties of the atmosphere. These procedures

already provided reasonably accurate proﬁles of the vertical ﬂux density and the

radiative cooling rate for various types of air masses as was veriﬁed by compari-

son with measurements. These methods required that the integration over the wave

number was carried out once and for all. Then followed the integration over the

inhomogeneous path of the atmosphere either using a graphical or a numerical pro-

cedure. This was done with the help of a one-parameter scaling technique which

will be discussed below. In order to improve the accuracy of the one-parameter pro-

cedure, the Curtis–Godson approximation was introduced which is a two-parameter

scaling technique. Higher order scaling techniques are possible but usually they are

quite difﬁcult to apply. In the following we will describe one- and two-parameter

models.

7.4.1 One-parameter scaling

The most simple scaling procedure is the so-called one-parameter scaling. Fortu-

nately, this method is sufﬁciently accurate to handle various transfer problems. The

transmission function for a non-homogeneous vertical atmospheric path may be

expressed by

T (u) =

ν



ν

exp



−



ν,L

(p, T )du



dν

ν



ν

exp



−





L,i

(ν − ν

0,i

)

+ α

L,i



dν

(7.219)

which applies to a small spectral interval containing numerous spectral lines. To

be speciﬁc we will assume that these lines have Lorentzian proﬁles, but for brevity

henceforth we omit the index L.

For simplicity let us ﬁrst consider the technique as it applies to a single spectral

line. The formal extension to include neighboring lines is simple. We have shown in

(7.27) that the Lorentzian half-width is proportional to the linear pressure reduction

(p/ p

) while the temperature dependency varies according to

√

/T . There is

sufﬁcient empirical and some theoretical evidence that the square root law to handle

the temperature dependency is not always sufﬁcient. Hence we will express the

temperature dependency by a more general law as stated by

α = α





(7.220)

7.4 Transmission in inhomogeneous atmospheres 257

leaving the exponent n unspeciﬁed. Since ( p

, T

) refer to the reference pressure and

temperature, the ratio of the absorption coefﬁcients k

(p, T )/k

, T

)isgiven

(p, T )

, T

)

S(T )

(ν − ν

)

+ α

S(T

)

(ν − ν

)

+ α

≈

S(T )

S(T

)





≈





(7.221)

Here, we have assumed that the strong-line approximation is sufﬁciently accurate

for the one-parameter scaling procedure. Thus, by ignoring the square of the half-

width in the denominator of the spectral line, the wave number-dependent parts of

the absorption coefﬁcient can be factored out and cancel. Furthermore, we have

assumed that the ratio S(T )/S(T

) is approximately equal to 1. Hence we may

write



(p, T )du = k

, T

)







du = k

, T

)

u (7.222)

where the quantity

u =







(7.223)

is known as the scaling parameter or the scaling path length. In case that there

are many spectral lines in the interval, the approximation used in (7.221) must be

replaced by

(p, T )

, T

)

≈



(T )α

0,i

(ν − ν

0,i

)







)α

0,i

(ν − ν

0,i

)

≈





(7.224)

However, the same scaling parameter

u is used whether we are scaling the absorption

path of a single line or a group of lines.

As stated above, most early radiative transfer calculations were carried out with

the help of radiation charts or the emissivity method. These methods were designed

to employ the scaling parameter

u which can also be successfully employed in

connection with the rotational water vapor band, see Chou and Arking (1980). In

the older literature the parameter

u was called the reduced absorber mass.

The extension to an inclined path, assuming horizontally homogeneous sublay-

ers, is quite simple as we have seen in our previous work. All we need to do is to

introduce the factor 1/µ in the exponent of (7.219).

258 Transmission in spectral lines and bands of lines

7.4.2 The two-parameter scaling technique of Curtis and Godson

To simplify the transmission calculations, the approximation replaces an inhomoge-

neous by a more or less equivalent homogeneous layer. This will be accomplished by

adjusting the parameters appearing in the absorption coefﬁcient which is assumed

to have the Lorentzian shape. The mean transmission in correct form is given by

(7.219). For simplicity let us again begin with a single spectral line by omitting the

summation sign. In the limit of the strong line approximation T

(u) the half-width

in the denominator may be ignored. Instead of neglecting α

altogether, we replace

it by a suitable mean value

which is independent of the atmospheric path

(u) =

ν



ν

exp



−



S(T )

α(p, T )

(ν − ν

)

+ ˜α



dν (7.225)

By expanding the exponent and discontinuing the expansion after the second term,

we obtain the weak line limit T

(u) of the transmittance. Since the interval ν is

assumed to be much larger than the half-width of the line, we may extend the wave

number integration to inﬁnity yielding

(u) =1 −

ν



S(T )α( p, T )

π ˜α



∞

−∞

+ 1

dx =1 −

ν



S(T )α( p, T )

˜α

(7.226)

Here we have used the simple transformation x = (ν − ν

a. In order to evaluate

a, we also expand the exponent of (7.219) and obtain the approximation

= 1 −

ν



S(T )du (7.227)

which, of course, is identical with the result obtained from the Ladenburg–Reiche

formula (7.73). Forcing agreement between (7.226) and (7.227) we obtain the

scaling half-width

˜α =



S(T )α( p, T )du



S(T )du

(7.228)

which is the ﬁrst scaling parameter of the Curtis–Godson approximation. By ignor-

ing the small temperature dependence of the half-width, that is

˜α

≈

(7.229)

we obtain the so-called pressure scaling factor

p =



S(T ) pd u



S(T )du

(7.230)

7.4 Transmission in inhomogeneous atmospheres 259

To simulate the transmission through an inhomogeneous atmosphere by an equiv-

alent homogeneous layer, using Goody’s (1964b) notation, we employ the adjusted

parameters

u. Thus, instead of (7.219) (omitting the summation sign), we

write

T =

ν



ν

exp



−

S ˜α

π[(ν − ν

)

+ ˜α

]



dν (7.231)

We now wish to obtain explicit expressions for

a and

u. Comparing (7.231) with

(7.225) and utilizing (7.228) gives

S ˜α

u =



Sαdu,

u =



Sd u,

u =



Sd u

(7.232)

For a moment one is tempted to conclude that we have a three-parameter approxi-

mation. However, only the parameters

a and

u are needed. The pressure variation of

the inhomogeneous atmosphere is included in (7.228) or (7.230) while the temper-

ature variation is modeled by (7.232). The mean line intensity

S can be evaluated

at any speciﬁed temperature since the effect of this temperature cancels. Thus the

two scaling parameters are

a (or

p) and

For a system of many lines the principle of the Curtis–Godson approximation

is the same as for a single line. We repeat equation (7.219) but we introduce the

line-shape factor of the Lorentzian line according to (7.18)

T (u) =

ν



ν

exp



−





f (ν − ν

0,i

,α

)du



dν (7.233)

In case of the weak line approximation this equation assumes the form

(u) = 1 −

ν







ν

f (ν − ν

0,i

,α

)dν du = 1 −

ν





dν

(7.234)

where we have assumed that the normalization condition (7.17) of the line-shape

factor is valid in the frequency interval ν. In order to introduce the strong line

approximation, as before, in the denominator of (7.219) we replace the half-width

by the constant

yielding

(u) =

ν



ν

exp



−





(ν − ν

0,i

)

+ ˜α



dν (7.235)

260 Transmission in spectral lines and bands of lines

To obtain the proper value for

a that ﬁts the strong line and the weak line approxi-

mation, we expand the exponent and ﬁnd

(u) = 1 −

ν





˜α



ν

f (ν − ν

0,i

, ˜α

)dν du = 1 −

ν





˜α

(7.236)

We proceed analogously to the single line case by comparing (7.234) and (7.236).

This gives





du =





du (7.237)

Since the pressure dependence of each line is identical, we obtain the ﬁrst scaling

parameter

p =



σ pd u



σ du

(7.238)

where σ = 1/N



is the mean line intensity. In case of a single line this equa-

tion, as it should, reduces to (7.230).

Now we need to ﬁnd the second scaling parameter. Let us reconsider equation

(7.219) whose exact analogy to a homogeneous path can be written as

T (u) =

ν



ν

exp



−



˜α



(ν − ν

0,i

)

+ ˜α





dν (7.239)

Comparison of (7.235) with (7.239) gives the second scaling parameter

u =





(ν − ν

0,i

)

+ ˜α



˜α

(ν − ν

0,i

)

+ ˜α

(7.240)

Recalling (7.228) and applying it to line i, after introducing the line-shape factors,

cf. (7.18), results in

u =



f (ν − ν

0,i

, ˜α

)





f (ν − ν

0,i

, ˜α

)

(7.241)

which is the scaled absorber mass. The line intensity

, as explained before, can

be evaluated at any speciﬁed temperature.

7.5 Results 261

The quality of the Curtis–Godson approximation for spectral intervals contain-

ing many spectral lines has been tested by Walshaw and Rodgers (1963). They

performed extensive cooling rate calculations using a line-by-line integration tech-

nique. For the water vapor rotational band and the 15

µmCO

band they found

errors less than a few percent. Various authors have investigated the accuracy of

the Curtis–Godson approximation. For example, Zdunkowski and Raymond (1970)

investigated the transmission in small spectral intervals in the 1.9 and 6.3

µm water

vapor bands. They found excellent agreement between the exact calculations and

the Curtis–Godson approximation. The Curtis–Godson approximation may also be

applied to other line shapes than the Lorentz line, but we omit any discussion.

While in many situations the Curtis–Godson approximation provides satisfactory

results for atmospheric water vapor and carbon dioxide distributions, the method is

less satisfactory for the 9.6

µm ozone band. Goody (1964b) employed van de Hulst’s

(1945) rather general technique to handle the transmission calculations pertaining

to inhomogeneous atmospheres. This technique is based on the series expansion of

the Fourier cosine transform permitting the formulation of three scaling parame-

ters. The extension of the Curtis–Godson scaling method resulted in a signiﬁcant

improvement of the transmission calculations. It is also possible to use additional

scaling parameters. This, however, greatly complicates the calculation procedure.

For more details the reader is referred to Goody’s original work.

7.5 Results

In this section we will brieﬂy consider some typical vertical proﬁles of solar and

infrared ﬂux densities as well as radiative temperature changes which have been

obtained by means of the radiation transfer model DISORT (Stamnes et al., 1988)

with a total of four discrete streams. To handle the spectrally dependent absorption

by atmospheric gases we have used the correlated k-distribution parameterization

of Fu and Liou (1992). Presently this appears to be the most efﬁcient yet sufﬁciently

accurate way to handle the spectral integration. The calculations specify the ground

albedo and the solar zenith angle as A

= 0.1 and θ

= 30

◦

for the solar spectrum

and a ground emissivity ε

= 1 for the entire thermal emission spectrum. Further-

more, the calculations assume clear sky conditions (no clouds and no aerosol) in

a mid-latitude summer atmosphere with typical vertical distributions of the radia-

tively relevant atmospheric trace gases. Figure 7.16 shows the vertical distributions

of temperature, relative humidity and those of the atmospheric trace gases which

have been used in the radiation calculations. The height constant CO

volume mix-

ing ratio has been set to 350 ppmv where 1 ppmv = 10

−6

. The integrated or total

amount of ozone in a vertical column above the surface of the Earth is expressed

in atmosphere centimeters (atm-cm) which is the height of the resulting volume

262 Transmission in spectral lines and bands of lines

220 240 260 280 300

Temperature (K)

Altitude (km)

0 20 40 60 80 100

Relative humidity (%)

Altitude (km)

−10

−8

−6

−4

−2

Volume mixing ratio

Altitude (km)

Fig. 7.16 Vertical distributions of temperature, relative humidity and the radia-

tively active trace gases for a mid-latitude summer atmosphere.

7.5 Results 263

0 200 400 600 800 1000 1200

Flux density (W m

−2

)

Altitude (km)

s,dir

s, +dir

i,i,

t, net

Fig. 7.17 Distribution of upward (↑), downward (↓) and net ﬂux densities labeled

as s (solar), i (infrared), t (total), s,dir (direct downward solar radiation) and net

(net radiation).

if all the ozone in the column of unit area were brought to normal pressure and

temperature (NTP). Currently, the total amount of atmospheric ozone is expressed

in terms of Dobson units (DU) where 1 DU = 10

−3

atm-cm NTP. The total ozone

column shown in Figure 7.16 amounts to 330 DU.

The absorption bands of the gases have been fully accounted for in the solar and in

the infrared spectrum. All transfer calculations pertaining to the solar spectrum use

a solar constant of 1368 W m

−2

while the infrared calculations assume a black-body

ground emission.

Figure 7.17 depicts the upward and downward directed solar and infrared ﬂux

densities and the corresponding net ﬂux density. The value of the upward solar ﬂux

density at the ground is found by multiplying the value of total downward solar

radiation reaching the ground (s,↓+dir) by A

= 0.1. The ﬂux densities at the

ground are needed to formulate energy boundary conditions in the thermodynamic

parts of weather and climate prediction models. Of particular interest is the radiative

ﬂux divergence of the combined solar and infrared spectrum, which is needed to

evaluate thermodynamic equations of the type (2.44).

Radiative heating rates corresponding to the radiative ﬂux densities of

Figure 7.17 are shown in Figure 7.18. While the vertical ﬂux density proﬁles are

rather smooth, radiative temperature changes vary quite rapidly with height. This

variation results from the vertical structure of the volume mixing ratio of the trace

264 Transmission in spectral lines and bands of lines

−10 −5 0 5 10 15 20

Heating rate (K day

−1

)

Altitude (km)

solar

infrared

net

Fig. 7.18 Solar heating rates, infrared cooling rates and the net radiative temper-

ature changes expressed in K day

−1

gases and from the atmospheric vertical temperature distribution. Inspection of the

curves shows that solar heating at the ground amounts to about2Kday

−1

. Had a

normal aerosol distribution been included, we would have calculated an additional

heating of about 0.1 K day

−1

. This is a small but not entirely negligible effect.

Since aerosol concentrations usually decrease with height, the aerosol contribution

to the heating rate is negligible at some height above the atmospheric boundary

layer. Solar heating, mainly caused by the presence of water vapor, decreases rather

rapidly with height in the troposphere due to the vertically decreasing water vapor

concentration. In the stratosphere strong solar heating is observed resulting from an

increase of the ozone volume mixing ratio with height. At a height of 42 km where

the ozone concentration begins to decrease, a maximum value of nearly 20 K day

−1

is obtained. For a smaller value of µ

maximum heating would have occurred at a

somewhat smaller height. For increasing values of µ

solar heating is most effective

in the upper part of the O

layer. As stands to reason, maximum heating will then

be less intense due to the smaller O

concentations existing there.

Let us now consider the infrared cooling rates which strongly depend on the con-

centration of the absorbing and emitting gases and also on the vertical temperature

distribution. For the midlatitude summer model, the tropospheric cooling rate of

2 K day

−1

, mostly due to water vapor absorption bands, is nearly height constant.

Additional calculations (not shown) reveal that larger tropospheric cooling will be

found in the moist tropical and smaller cooling in arctic air masses. Moreover, the

cooling rates will be strongly modiﬁed by the presence of clouds.