Salby M.L. Fundamentals of Atmospheric Physics

220

E

o

v

z

o

m

i-.

o

iJJ

o3

0

cc

o

z

0

l-

n

0c:

0

o)

n7

<

8 Atmospheric Radiation

ABSORPTION BY

0 2

1016

1017

10

TM

1019

10 ,0

1021

1622

10:'3

1024

1625

m

w

B

m

I

5O

CONTINUUM

ION I

c o.t,.uuM

1 '

LYMAN o~

11.

I I I I I I I I I I I I I

100 150

WAVELENGTH (nm)

HERZBERG

CONTINUUM

m

I I I I I I

200 25O

Figure 8.12 Absorption cross section as a function of wavelength for molecular oxygen. After

Brasseur and Solomon

(1986).

Reprinted by permission of Kluwer Academic Publishers.

of 03, whereas photodissociation of 02 dominates at shorter wavelengths. Ow-

ing to the inverse relationship between wavelength and penetration altitude

(Fig. 8.3), these bands influence different levels. Absorption by 03 in the

Hartley and Huggins bands at A > 200 nm dominates in the stratosphere and

mesosphere, where it provides the primary source of heating. Ozone absorp-

tion in the Chappuis band at A > 400 nm becomes important below 25 km.

On the other hand, absorption by 02 at A < 200 nm prevails only only above

60 km. Despite its-secondary contribution to absorption, photodissociation of

02 plays a key role in the energetics of the stratosphere and mesosphere be-

cause it produces atomic oxygen, which supports ozone formation (1.27). In

addition to molecular oxygen and ozone, water vapor, methane, nitrous oxide,

and CFCs also absorb in the UV through photodissociation.

8.3.2 Line Broadening

Absorption lines in Figs. 8.11 and 8.13 are not truly discrete; rather, they

occupy finite bands of wavelength due to practical considerations surrounding

molecular absorption and emission. The spectral width of an absorbing line is

described in terms of a shape factor. The absorption cross section at frequency

~, is expressed

O'a,, -- Sf(v-

v0), (8.30.1)

8.3 Absorption Characteristics of Gases

SCHUMANN-RUNGE BANDS

221

r

E

v

c-

O

,i

,,I,-,*

~d

O9

r

0

t_

0

c-

O

0

0')

..Q

.<

I

19-o 17-o

15-_o

_,~L[ ' I

10 1 i

1619

10-2~ i

56500

1(~21 18-0 A 16-0 I

57000

i

10181r 17-1

I" 8-0

I 11-

1619 . l

16 20

0-2

i '55

1 12511d0

I~00 0

14 -0

7-0

I-

Io-i

13-0

I :~---

i

56000

I

]

'

1

16~[~II 7 ~ 14-~ 3-0

" 6-1~ 2-0

1621 ,

I, r--llllJ, s-1 l 4-I 1-o

r-- l, [ 3-,

10 52000

"!

!

[

2-1 !

F---

51000 50000 49000

Wavenumber (cm "1 )

Figure 8.13 Absorption cross section in the Schumann-Runge absorption bands of molecular

oxygen. Shown at high spectral resolution. After Kocharts (1971). Reprinted by permission of

Kluwer Academic Publishers.

where

s

S = J ~r~dv (8.30.2)

is the

line strength,

v 0 is

the line center, and the shape factor f accounts for

line broadening.

222 8

Atmospheric

Radiation

ABSORPTION BY

0 3

-17

04

E

-18

Z

o

-19

o

uJ

03

03 -20

O

rr

o

o -21

(.5

o

.J

INS

-22

/~-"/ C HAPPUIS ~~~_.

/

-23 ' ' ~ ' ' ,

t t , J , , ~) J

200 300 400 500 600 700 8 0

WAVELENGTH (nm)

Figure 8.14 Absorption cross section as a function of wavelength for ozone. Sources: Andrews

et aL

(1987) and WMO (1986).

WAVELENGTH (nm)

833 526 385 303 250 213 185 164

-17.6 ~ I I I I I i

-18.8

04

E

o

z -20.0

O.--.

o

w

co ~ -21.2

O~

rr _

o ~ -22.4

Z W

r- 3

a. -22.E

0

m

.<. -24.8

0

._1

-26.(1 ~,

~o2X 0.21

~%x 10 -5

I

2.6 3.3 4.0 4.7 5.4 6.1

WAVENUMBER x 10 -4 (cm -1)

Figure 8.15 Relative contributions to absorption cross section from ozone and molecular oxy-

gen. Adapted from Brasseur and Solomon (1986). Reprinted by permission of Kluwer Academic

Publishers.

8.3 Absorption Characteristics of Gases

223

A fundamental source of line width is

natural broadening,

which results from

the finite lifetime of excited states. Perturbations to the radiation stream then

introduce a natural width that has the

Lorentz line shape

where

fL( - =

7T(P -- /20)2 + O~2L ' (8.31.1)

a L = (2rr~)-1 (8.31.2)

is the half-width at half-power of the line (Fig. 8.16) and t is the mean lifetime

of the excited state. For vibrational and rotational transitions in the IR, natural

broadening is insignificant compared to other sources of line width. The two

most important stem from the random motion of molecules and collisions

between them.

Collisional

or

pressure broadening

results from perturbations to the absorb-

ing or emitting molecules, which are introduced through encounters with other

molecules and which destroy the phase coherence of radiation. The spectral

width introduced by collision is modeled in the Lorentz line shape (8.31.1),

but with the collisional half-width

1

ac = a0 -~- , (8.32)

which, from kinetic theory, is inversely proportional to the mean time between

collisions and where a 0 ~ 0.1 cm -1 is the half-width at standard temperature

and pressure T 0, P0.

F-

z

I.iJ

0

I J_

1.1_

LH

0

Z

0

Q..

rr

0

03

133

<

f\

/ \

Lorentz

/ \

/ ....

Doppler

/ 9

/." '~

Voigt

........ / ": ../

/"" "1

/..i~ I

ii .. "1

.'i

.'" /

'"'" - "-" ~

I

"- "-- - - '

v 0

Figure 8.16 Lorentz, Doppler, and Voigt shape factors describing finite spectral line width.

Reconstructed from Andrews

et

al. (1987).

224 8 Atmospheric Radiation

Molecular motion v along the line of sight introduces another source of

line width.

Doppler broadening

follows from the frequency shift

( v) (8.33.1)

V-Vo 1+ c

of individual molecules undergoing emission or absorption. In combination

with the Boltzmann probability distribution

}/ m exp(my2)

(8.33.2)

p(v) - 27rKT - 2KT '

where m is the molecular mass, (8.33.1) leads to the shape factor

1 [ l

fD (v -- v o) -- aD ~

exp

-- aD

, (8.34.2)

where

Vo /2KT

(8.34.2)

aD ~

~Vc m

is the Doppler half-width divided by a factor of ~ 2. The Doppler line shape

is illustrated in Fig. 8.16 along with the collisional line shape for

aD = ac.

Unlike the collisional half-width, aD does not depend on pressure.

Below 30 km, the pressure dependence in

ac

makes collisional broadening

dominant for IR bands of CO2 and

H20.

At higher altitudes, Doppler and

natural broadening become significant at wavelengths in the visible and UV.

Natural broadening of the Schumann-Runge bands of O2 becomes comparable

to collisional broadening in the upper stratosphere and mesosphere. Those

two sources of line width are treated jointly in the

Voigt line shape,

which is

superposed in Fig. 8.16. When they are comparable, the Doppler profile is

dominant near the line center, whereas the Lorentz profile prevails in the

wings of the line. The Voigt profile gives a line shape between the two.

8.4 Radiative Transfer in a Plane Parallel Atmosphere

In a stratified atmosphere, properties vary sharply with height. It is therefore

convenient to treat radiative transfer within the framework of a

plane parallel

atmosphere,

in which

9 Curvature associated with sphericity of the Earth is ignored.

9 The medium is regarded as horizontally homogeneous and the radia-

tion field horizontally isotropic.

Then, along a slant path

ds

(Fig. 8.17), a pencil of radiation inclined from

the vertical at a zenith angle 0 traverses an atmospheric slab of thickness

dz = tzds,

(8.35.1)

8.4

Radiative Transfer in a Plane Parallel Atmosphere

225

r

0

zz (z)

"~s;L

Figure

8.17 Plane parallel atmosphere, in which a pencil of radiation is inclined at the zenith

angle 0 = cos -1 /z. Elevation is measured by the optical depth for a given wavelength ~-x, which in-

creases downward from zero at the top of the atmosphere to a surface value of ~'sx.

where

/z = cos 0. (8.35.2)

Introducing the

optical depth

f

OO

~'A -- pkAdz'

1 P

- g fo kAdp'

-/zXA, (8.36)

which is measured downward from the top of the atmosphere, transforms the

budget for a pencil of radiation (8.28) into

d/A

/z~-~-A - I,~ - J,~, (8.37)

where I~ - IA(~- ~,/z) and JA = J~(~A,/x). Proceeding as in the treatment of

(8.28) leads to a formal expression for upwelling radiation (0 < /z <_ 1) in

226 8 Atmospheric Radiation

terms of that at the surface (r a =

rsa),

Ia(z' l'~) = Ix(zsx' lx)exp [ ra(z) - 'rsx

+ JA(z~,/x) exp

a~A(z) lz IZ

0</z_< 1,

(8.38.1)

and for downwelling radiation (-1 </z < 0) in terms of that at the top of the

atmosphere (r~ = 0),

Ia(z' tz) = Ia(O' tz)exp [ ra(z)

_

f'~(z)ja(rx, tz)exp rx(z)

9 10 i ~

-l_</z<O.

'] dr~

T h

/z

(8.38.2)

As before, the first terms on the right-hand sides of (8.38) describe the extinc-

tion of radiation incident at the boundaries, whereas the second terms account

for the cumulative emission, absorption, and scattering into the pencil between

the boundaries and the height z. If Ia is known at the upper and lower bound-

aries and if Ja can be specified in terms of known properties, (8.38) determines

the radiation field throughout the atmosphere. Alternatively, if Jx, p, and k

can all be specified in terms of temperature [e.g., in the absence of scattering

(8.18) and under hydrostatic equilibrium (1.17)], then (8.38) determines the

thermal structure of the atmosphere. The expressions for the radiation field

then provide integral equations for the temperature distribution

T(z).

Their

solution defines the

radiative-equilibrium thermal structure,

which corresponds

to a balance between radiative components of the energy budget.

Horizontal homogeneity and isotropy make the upwelling and downwelling

fluxes

fo

F~(zA)

--

2Ir IA(rx,/.Q/~d/.~

f_o

F1(ra) = -27r I~(r A,/x)/xd/x

1

(8.39)

in (8.1) and (8.3) the essential descriptors of radiative transfer in a plane

parallel atmosphere. By integrating (8.38) over zenith angle and incorporating

8.4 Radiative Transfer in a Plane Parallel Atmosphere

227

(8.4), we obtain

f01 ( TA )

F~a (%) -- 2rr Ia(rsa, /,)exp - rsa tzdl~

tx

f%afO1 t

(TA -- TAt)

+ 2rr Ja(r a,/x) exp

dlzdr' a ,

J T A J~

F+a(ra)

- -2rr Ix(0,/,1 exp b~eb~

1

fo, f_ o , (.-.,)

+ 27r JA(%, Ix)

exp

dtzdr' a.

1

tz

(8.40.1)

(8.40.2)

For LW radiation and in the absence of scattering, the source function is just

Ba(T).

Upwelling radiation at the surface, which is taken to be black, is then

given by

I,(rsa,

b~)= Ba(Ts) 0 < ~ < 1. (8.41.1)

Likewise, downwelling LW radiation at the top of the atmosphere must vanish,

SO

Ix(0,/x) = 0 - 1 </, < 0. (8.41.2)

The upwelling and downwelling fluxes of LW radiation can then be written

f01 ( TA )

Fa~(ra) -

27rBa(Ts)

exp - rsa ~db~

frsaf01 (t)

+ 2 ~rBa[T(r'a) ]

exp

ra - ra dlxdr,a,

,1 g A I ~

fofo 1 (l)

F+a (%) = 2 ~" rrBa[T(r'a)]ex p ra - r~ dl~dr,a.

IX

(8.42.1)

(8.42.2)

The integrations over zenith angle may be expressed in terms of the expo-

nential integral (Abramowitz and Stegun, 1972)

f

oo e-Xr

En(T ) =

--~-dx,

(8.43.1)

which satisfies

dEn

dr

= --En_I(T ). (8.43.2)

228 s

Atmospheric

Radiation

Letting x =/x -1 transforms (8.42) into

F~(Tx) =

2.n'Bx(Ts)E3('rsx -

rx)

f TsA

+ 2 ,n'B,~[T(,r'~)]EE('r ~ - "rx)d'r~,

(8.44.1)

o, T A

F~ ('r;t ) - 2 fo ~'~ "n'B;t[ T('r~)]E2('rx - "r'x)d'r ~.

(8.44.2)

Then integrating over wavelength recovers the total upward and downward

fluxes

fo

Ft('r) - 2 "rrB;L(Ts)Ea('rs, ~ -

~'a)dh

+ 2 J~ Jo

=B.[r(,i)IE2( k

-

,r,~)dhd,r',

(8.45.1)

fo fo

F~'('r) = 2 ,n'B;t[T(,rl)]E2(z,~ - r',x)dhd'r' ,

(8.45.2)

which constitute the formal description of LW radiative transfer in a nonscat-

tering atmosphere.

8.4.1 Transmission Function

Implementing (8.45) is complicated by the rapid variation with wavenumber

of absorption cross section, which involves thousands of vibrational and rota-

tional lines in the IR. In place of line-by-line calculations, radiative transfer is

calculated with the aid of band-averaged transmission functions that embody

the general characteristics of the absorption spectrum in designated ranges of

wavelength.

Integrating (8.44) over a frequency interval Av yields the band-averaged

fluxes

F~r (z~) - 27rBv(T~) fa E3(z~, - z~)a,d--~

+ 2 E2(G -

J T b l~

F~(~'v) - 2 7rB~[T(~',~)] Ez(z,, - r eL,,

P

(8.46.1)

where B~ can be used for the band-averaged emission because the Planck

function varies smoothly with frequency. The

band transmissivity

or

transmis-

sion function

is defined as

f~ "/'v

~(~'~,/x) = ~ ~ e-; dr.

(8.47)

(8.46.2)

8.4 Radiative Transfer in a Plane Parallel Atmosphere 229

Decreasing downward, ~ represents the fractional intensity in the band and

in direction/z that reaches optical depth z~. The

band absorptivity

is then

a~ = 1 - ~, (8.48)

analogous to (8.11).

Averaging (8.47) over zenith angle defines the

diffuse flux transmission func-

tion

for the band

~(r~) = fo

,t

/~)/.,d/.,

fo 1/xd/x

f0

1

= 2 ~(z~,/z)/zd~, (8.49)

which can be written in terms of the exponential integral as

f~ dv (8.50/

~-~(~'~) = 2 E3(~'~) A---v"

With the aid of (8.43.2), the band-averaged fluxes can then be expressed in

terms of the flux transmission function

F}(z~) = ~rB~(T~)~(%~ - ~)

-

- 7rn~[r(r'~)] dr"

dr'~ ,

(8.51.11

f0 "~' dr (8.51.2)

d~(%

g'v)

Fr (r = + ~rB~ [T(r dr

In principle, absorption characteristics of the band Av determine the corre-

sponding transmission function and the vertical fluxes through (8.511. Collect-

ing contributions from all absorbing bands then obtains the total upwelling and

downwelling fluxes. However, in practice, even individual absorption bands are

so complex as to make direct calculations impractical (refer to Figs. 8.11, and

8.13). Instead, the transmission function is evaluated with the aid of band

models that capture the salient features of the absorption spectrum in terms

of properties like mean line strength, line spacing, and line width. The regular

band model of Elsasser (1938) treats a series of evenly spaced Lorentz lines,

like those comprising the CO2 14-/zm band (Fig. 8.118). The Goody (1952)

random model treats a spectrum of lines that are randomly spaced, as is typ-

ical of the 6.3-/.~m band of water vapor (Fig. 8.11b). See Liou (1980) for a

development of these models.

The transmission function, although more complex than the exponential

attenuation of monochromatic radiation (8.9), is far simpler than a line-by-line

calculation over the band. It is instructive to consider how the transmission

function, or alternatively the absorptivity (8.48), behaves with optical depth for

an individual absorption line. Below a certain height, the absorptivity of a line

Salby M.L. Fundamentals of Atmospheric Physics

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