2-1

CChhaapptteerr 22

2-1 2-1-1

c ~= (2.1)

c 1~ == (2.2) (wave length) ; ~ (frequency)

(wave number) ; c (= 2.99793*108m sec-1) m 10-6 m () (micro-meter) ; nm10-9 m (nano-meter)

()10-10 m

UL ; IR

alpha beta gamma delta epsilon zeta eta theta kappa lambda mu /mju/ nu /nu/ or /nju/ xi /ksi/ rho sigma tau phi chi psi omega

Fig 2.1: The electromagnetic spectrum in

terms of wavelength in m , frequency in GHz, and wavenumber in cm

-1. [Liou02,

Figure1.1]

2-2

2-1-2 solid angle

2r= (2.3)

=

= 24 r

= 4

Fig2.3 )sin)(( drdrd = (2.4) ddrdd sin2 == (2.5)

Fig 2.2: Definition of a solid angle , where denotes the area and r is the distance. [Liou02, Figure1.2]

Fig 2.3: Illustration of a differential

solid angle and its representation in

polar coordinates. Also shown for

demonstrative purposes is a pencil of

radiation through an element of area dAin directions confined to an element of

solid angle d . Other notations defined in the text. [Liou02, Figure1.3]

2-3

2-1-3

(dA) d()(dt)

+d dtddAdIdE = cos (2.6)

dAdtdddEI

=

cos)( (2.7)

() Defining (F) (() )

= dIF cos (2.8)

=

dAdtddE

(2.5)

( ) =

ddIF2

0 0

2 sincos,

(2.9)

( ) ,I (isotropic radiation)

=

ddIF2

0 0

2 sincos

(2.10)

I = (2.11) Defining (F) (() )

=

0dFF (2.12)

Defining ()(f) (flux) ( = w )

= AFdAf (2.13)

Table 2.1 [Liou02, Table1.1]

2-4

2-1-4 () (Scattering)

( ax 2= )

Rayleigh scattering () 1x (geometric optics)

Figure 2.5: Multiple scattering process

involving first(P), second(Q), and third (R) order scattering in the direction denote by d.[Liou02 ,Figure1.5]

Fig 2.4: Demonstrative angular patterns of

the scattered intensity from spherical

aerosols of three sizes illuminated by the

visible light of 0.5 m ,(a) 10-4 m , (b) 0.1 m , and (c) 1 m . The forward scattering pattern for the 1 m aerosol is extremely large and is scaled for

presentation purposes. [Liou02, Figure1.4]

2-5

2-1-5 (extinction cross section) (extinction coefficient)

(m2)

0Ff = ( see (2.13) )

0F

f

n = (m-1)

n 2-2

(blackbody radiation):

()

(homogeneous)(unpolarized)(isotropic)

2.6

2-2-1 Plancks ()

nhE ~= (2.14)

)1(2

5

2

= Thce

hcB (2.15)

BPlancks ( I)

h Plancks = 6.626810-34 Jsec

Boltzmann = 1.3806810-23 Jdeg-1

~ n

depends on T() ()(2.15)

Figure 2.6: A blackbody radiation cavity to

illustrate that absorption is

complete.[Liou02,Figure1.6]

2-6

Tm

310*897.2 =

2-2-2 Stefan-Boltzmann -

40 )()( bTdTBTB ==

(2.16)

3244 15/2 hckb =

(isotropic) (see (2.11))

( ) 4)( TTB == (2.17)

(Stefan-Boltamann ) = 5.67*10-8 Jm-2

(T)

2-2-3 Wiens -

0)( =

TB (2.18)

(2.19)

2-2-4 Kirchhoffs

(absorption)(emission) ( ) ( ) tyabsorptiviAemissivity = (2.20)

=

Fig 2.7: Blackbody

intensity (Planck

function) as a

function of

wavelength for a

number of emitting

temperatures.

[Liou02, Figure1.7]

2-7

=

1== (2.21)

1

2-8

(/ = = )

dsJkdI =

Combining (2.23), (2.24) and (2.25)

dsjdsIkdI += (2.26)

dsJkdsIk += (2.27)

JIdsk

dI+= (2.28)

NoteWhen radiation passes through a medium, the change of radiation is determined

by

I extinction of incident radiation, including absorption and scattering by

medium

J radiation sources in medium, including emission and multiple scattering

(2.28)

() ds (k)()

(ds)(I)

(J)

Fig 2.8: Depletion of the radiant intensity in traversing an extinction medium.[Liou02, Figure1.13]

2-9

2-3-2 Beer-Bouguer-Lambert (Beers) Assuming I(0) at s = 0, I(s1) at s = s1

From (2.23)

dskI

dI

=

( )

( ) =

11

00

1 s

sI

I

dskdII

( )

( ) =11

0 0ln

s

I

IdskI s

( )( )

=

1

01 exp

0s

dskI

sI

( ) ( )

=

1

01exp0

s

dskIsI

(2.29)

If the medium is homogeneous, defining (path length)

=1

0

sdsu

uk

eIsI = )0()( 1

the change of the radiation exponentially depends on the path length of the

medium.

Defining (optical depth) ()

( ) 1

010,

s

dsks

(2.30)

(note ( ) dskssd =,1 )

( ) ( ) ( )0,1 10 s eIsI = (2.31)

( )

Note s ( 0) dskssd ),( 1 =

Defining (transmissivity)

( )( )0

I 1

Is

=

2-10

( )0,1se= (2.32)

For () ( )0,111 s eA

== (2.33)

For

1=++ RA (2.34)

R (reflectivity)

Note

2-3-3 Schwarzschild (2.38)

Considering (e.g.) Planck

( )( )TB J in(2.28) using(2.30) ( )( ) ( ) ( )( )sTBsIssdsdI

+= ,1

(2.35)

(2.35) ( )sse ,1

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )ssdesTBssdesIsdIe ssssss ,, 1,1,, 111 = ( ) ( ) ( )( ) ( ) ( )ssdesTBdesI ssss ,1,, 11 =

( ) ( ) ( ) ( ) ( )( ) ( ) ( )ssdesTBdesIsdIe ssssss ,1,,, 111 =+

( ) ( )( ) ( )( ) ( ) ( )ssdesTBsIed ssss ,1,, 11 = (2.36)

(2.36) from 0 to s1

( ) ( )( ) ( )( ) ( ) ( ) = 1 1 110 0 1,, ,

s s

ss

ss ssdesTBsIed

(2.37)

( ) ( ) ( )( ) ( ) ( ) =

11

11

0 1,

0

, ,s

ss

s

ss ssdesTBsIe

Fig 2.9: Configuration of the optical thickness defined in Eq. (1.4.15). [Liou02, Figure1.14]

2-11

( ) ( ) ( ) ( )( ) ( ) ( ) =+1

11

0 1,0,

1 ,0s

ss

s

ssdesTBeIsI

( ) ( ) ( ) ( )( ) ( ) ( ) =1

11

0 1,0,

1 ,0s

ss

s

ssdesTBeIsI

(2.38)

(2.28) 2-3-4

point ( ) ,;z ()

(2.28) ( ) ( ) ( )zJzIdskzdI ,;,;,; +=

( ) ( ) ( )zJzIdzkzdI ,;,;

sec,;

+=

( ) ( ) ( )zJzIdzkzdI ,;,;,;cos += (2.39)

Defining (normal optical depth)

=

zzdk ( dzkd = )

( )0= z

z

(2.39) ( ) ( ) ( )JIddI ,;,;,; = (2.40)

cos=

Following section 2-3-3 (2.40) e

( )0> ( from to * )

Fig 2.10: Geometry for plane-parallel

atmospheres where and denote the zenith and azimuthal angles,

respectively, and s represents the

position vector. [Liou02, Figure1.15]

Fig 2.11:

Upward ( ) and downward ( ) intensities at a given level and at top ( 0= ) and bottom ( = ) levels in a finite,

plane-parallel atmosphere.

[Liou02, Figure1.16]

2-12

( ) ( ) ( ) ( ) ( )

+= ** ,;,;,; *

deJeII

(2.41)

( )0> ( from 0 to ) ( ) ( ) ( ) ( )

+=

deJeII

0,;,;0,; (2.42)

(2.28) 1st term of r.h.s. (outside of to ) 2end term of r.h.s level of

(homogenous)

2-4

Fig 2.12(a) Blackbody radiation curves for temperature characteristic of the Suns surface

and upper levels on Earths atmosphere from which infrared radiation escape to space.(b)

Absorption of radiation at each wavelength if the beam passes through the entire atmosphere

from top to ground level or vice versa. (c) As in b, but for radiation passing from the top

of the atmosphere to 11 km or vice versa. After Trenberth (1992) and Goody and Yung (1989).

2-13

2.4-1 () 2.4-2 ()

Fig 2.13: Solar irradiance curve for a 50 cm

-1spectral interval at the top of the atmosphere (see

Fig.2.9) and at the surface for a solar zenith angle of 60oin an atmosphere without aerosols or

clouds. Absorption and scattering regions are indicated. See also Table 3.3 for the absorption

of N2O, CH4, CO, and NO2. [Liou02, Figure3.9]

Fig 2.14: Theoretical Planck radiance curves for a number of the earths atmospheric

temperatures as a function of wavenumber and wavelength. Also shown is a thermal infrared

emission spectrum observed from the Nimbus 4 satellite based on an infrared interferometer

spectrometer. [Liou02, Figure4.1]