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12.815 Atmospheric Radiation
Fall 2008
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12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
1. Equation of Radiative Transfer
Specific Intensity of Radiation
IQ T, I, z EQ Energy dA Area d: Solid Angle dQ frequency int erval dt time In an interval ds, we lose intensity by extinction (scattering and absorption) and gain it by
emission and scattering.
Lambert’s Law: The extinction process is linear, independently in the intensity of radiation and
in the amount of matter, provided that the physical state (i.e. temperature, pressure,
composition) is held constant.
From Lambert’s Law, the change of intensity along a path ds is proportional to the amount of
matter in the path and to the intensity of radiation:
dIQ extinction losses D Q IQ ds
where D Q = volume extinction coefficient
(1)
The argument that the extinction process is linear in the amount of matter applies with equal
force to the emission process. Therefore, we write:
dIQ gains D Q JQ ds
(2)
where we have defined the source function, JQ .
The extinction coefficient can be expressed as the sum of an absorption coefficient k Q and a
scattering coefficient VQ .
DQ
k Q VQ
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
(3)
Lecture
Page 1 of 15
The most general problem in atmospheric radiation, therefore, has a source function consisting
of two parts,
D Q JQ k Q JQ thermal V Q JQ scattering
(4)
where k Q JQ
HQ
4S
H Q
thermal
for isotropic emission.
VQ JQ scat , on the other hand, is given by two terms, describing the diffusely scattered
radiation and the singly scattered incident beam of radiation (the sun).
dIQ
T, I, z ds
D Q I T, I, z HlQ T, I, z VQ ³ ³ P T, I, T, ' I ' I T', I,' z
VQ
d4:S
(5)
SF
exp D Q z cos T0 P T, I, T0 , I0 4S
where P T, I, T,' I ' is the scattering phase function (or scattering diagram) and is normalized
d:
such that
³ P 4S
1 where d: is an element of solid angle. The shape of the phase function can
4S
d:
be usefully characterized by a single number, cos K !
³ (cos K)p 4S
where K is the scattering
4S
angle and <cos K > is called the asymmetry parameter (which varies between 1 and -1 and is 0
for isotopic scattering.
Dividing by D Q z , we have:
dIQ
T, I, z D Q z ds
1
IQ T, I, z H Q z DQ z VQ z 4S D Q z ³ ³
P T, I, T, ' I ' I T', I,' z sin T ' dT ' dI '
(6)
VQ z SF
D Q z 4S
exp D Q z z cos T0 P T, I, T0 , I0 Let us now introduce the following definitions:
dW Q
D Q dz
ds
dz
P
P
cos T
where WQ = vertical optical depth measured from the top of the atmosphere WQ
(7)
0 at z
f .
Note that this coordinate differs for each Q .
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 2 of 15
We then obtain the following Radiative Transfer Equation in differential form in a “plane
parallel” atmosphere:
dI WQ , P, I H Q WQ VQ WQ I WQ , P, I P T, I, T' , I' I WQ , T' , I' sin T' dT' d)
P
dWQ
D Q WQ 4SD Q WQ ³ ³
V Q WQ SF
4SD Q WQ exp D Q WQ z cos T0 P T, I, T0 , I0 (8)
or
P
dI WQ , P, I I WQ , P, I J WQ , P, I dW Q
where we have defined the source function, J WQ , P, I as:
J WQ , P, I iQ
H Q
Z
D Q 4S
³ ³ P P, I, P , I I W , P, I, P , I dP dI
'
'
'
'
'
'
Q
(9)
iQ SF
Z
Q
exp WQ P0 P P, I, P0 , I0 4S
and
VQ
Q = Single Scattering Albedo.
= Z
DQ
Let us also be reminded that the atmosphere in general contains both gases and particulates
(aerosols). Each has scattering and absorption properties that we need to consider. Thus we
have:
DQ
k Q gases k Q aerosols VQ gases VQ aerosols
There are many cases where the physical situation enables the neglect of one or more of these
terms and a related simplification of the Radiative Transfer Equation. We will now examine a
few such cases.
Let us first apply Kirchoff’s Law to our Radiative Transfer Equation. “The ratio of emission and
fractional absorption in any direction of a slab of any thickness in thermodynamic equilibrium
equals the black body intensity.” So – in a non-scattering atmosphere, we have
H Q T, I, WQ H Q WQ 4S
DQ
DQ
BQ T B Q WQ 2hQ3
1
c2 ehQ kT 1
Black Body Function
BO T 2hc2
1
O5 ehc OkT 1
where BQ WQ is isotropic
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 3 of 15
This gives us:
JQ WQ , P, I B Q WQ Q
Z
P P, I, P' , I' I P, I, P' , I' , WQ dP dI
4S ³ ³
(9a)
SF
Z
W
Q exp §¨ Q ·¸ P P, I, P0 , I0 4S
© P0 ¹
Q ,BQ WQ ) are horizontally invariant, then I is independent of
Also if all relevant properties ( D Q , Z
azimuth angle, I i.e. IQ WQ , P, I IQ WQ , P and we have:
P
dIQ P, WQ IQ P, WQ J WQ dW Q
(10)
Case I:
Let us first consider the simplest case characterized as follows:
a) We observe sunlight of visible wavelengths through a non-scattering atmosphere:
SF
and also BQ WQ IQ P, WQ 4S
BQ WQ or
b) Bright, artificial source of radiation shining through an atmospheric path (e.g.- laser):
We then have that IQ P, WQ !! JQ WQ and the Radiative Transfer
Equation is simplified to:
P
dIQ P, WQ dW Q
IQ P, WQ (11)
and the solution of this equation is:
I
³
Io
dIQ
IQ
WQ
³
0
dWQ
or IQ P0 , WQ P0
and since P0
IQ P0 , WQ W
I0Q P0 , 0 exp §¨ Q ·¸
© P0 ¹
cos T0 , we have
W
º
I0 Q P0 , 0 exp ª« Q
¬ cos T0 ¼»
Beer’s
Law
Bouguer’s Law
Lambert’s Law
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 4 of 15
If we have I(To, 0) as a slowly varying function of frequency, Q , and
We define the transmission function, t
³ I P, W dQ
Q
ª WQ
I T0 , 0 ³ exp «¬
'Q
'Q
W
º
exp ª« Q
¬ cos T0 »¼
º dQ
cos T0 »¼
I To , 0 t 'Q
A lot of work has been done to develop methods for computing this mean transmission and we
will return to this topic and examine it in detail a little later.
Case II:
Let us consider a cloudless atmosphere and the infrared portion of the electromagnetic
spectrum. Due to the approximate separation of the solar emission spectrum and planetary
emission spectrum (as discussed previously by Prof. Seager), we now have:
J WQ # B Q, WQ And the Radiative Transfer Equation reduces to:
P
dI P, WQ I P, WQ B P, WQ dW Q
(12)
This is a linear first order equation. If we apply eW P as an integrating factor, we obtain the
following equation:
eW P
dI
eW P
I
dW
P
B W P
e
P
d ª¬IeW P º¼
(13)
dW
Lets consider the upward intensity at a level, z (P > 0). The origin of optical depth is at the top
of the atmosphere and we will need to integrate from the level, z, to the surface. It is therefore
convenient to change the variable of integration to W ' W Wz as we must integrate over optical
depths ranging from zero to the optical depth at the surface of the earth. Thus, we have Eq.
14:
-τ '
Ιe
μ
τs
τz
=
τs
∫
τz
−B(τ ') -τ
e
μ
'
μ
dτ' =
τs
∫
τz
-B(τ) −(τ−τz ) / μ
dτ
e
μ
(14)
and finally the solution we are seeking:
I Wz I Ws e
Ws Wz P
Ws
³ B W e
WWz P
Wz
dW
P
(15)
where the subscript, s, in Eqs. 14 and 15 refers to the surface of the earth.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 5 of 15
And, at the ground we typically have nearly black body emission in the infrared, so I( Ws ) can be
replaced by H B Ws where H is an emissivity (near unity) and B Ws is the Planck Black Body
Function.
And the downward solution is similarly given by:
Wz
I Wz p
³ B W e
Wz W P
0
dW
P
(16)
In these equations we’ve dropped the frequency specification for simplification. But – we’ll need
to keep in mind that optical depth, Planck function and radiation intensity always depend on
frequency (or wavelength).
If the temperature is known throughout the atmosphere, an exact solution is possible: i.e. –
B W B T p Consider pressure coordinates for which
dW
D dz
k Ua dz
k Fa dp
g
k F a U dz
dp
§
¨ sin ce dz
©
·
Ug ¸
¹
where k Q is the absorption cross-section (in cm2/gm), Ua is the absorber mass density and Fa
is mass mixing ratio of absorber, a.
Now, lets go back to the formal solution of the R.T.Eq. (Eq. 15) and examine the details:
B W Ÿ B Tp e
Ws Wz P
ps
Ÿ exp ³
pz
p
e
Wz W P
Ÿ exp ³
pz
k Fa
dP
gP
{ Transmission
k Fa
dP
gP
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 6 of 15
and if the ground can be considered black (a reasonable assumption in much of the thermal
infrared), we have H B Q, T Ÿ B Q, Tg So – finally, we have:
I P,Pz n B Tg exp ps
³
pz
ps
k Fa
dp ground contribution
pz g P
³
k Fa
B Tp exp gP
ª p k Fa
º
dp '» dp atmospheric contribution
«³
«¬pz g P
»¼
(17)
For downward radiation I P, W p , we similarly have:
pz
I P,pz p
³
0
ªpz k Fa
º
k Fa
B Tp exp « ³
dp '» dp
gP
«¬ p g P
»¼
(18)
Let us recall that we defined the transmission function as follows:
t
ª p k Fa
º
exp « ³
dp »
»¼
¬«pz g P
From above, we have
tf
I P,pz p
³ B Tp dt
(19)
1
where T = T(t)
with a similar expression for the upward intensity:
ts
I P,pz n
³ B T dt B Ts t s
(20)
1
where t = transmission
In practice molecular absorption by atmospheric gases (H2O, CO2, O3, N2O, CO, CH4, O2, etc.)
fluctuate rapidly with respect to frequency compared with the Planck function B Q, T .
Therefore, when we consider a spectral interval appropriate for measurement, we can take the
frequency variation into account and we have:
tf
I'Q p
³
'Q
where t
IQ P,pz dQ
³ B Q, T dt
1
p
°­ z k Q Fa dp ' ½°
exp
®
³
³ g P °¾ dQ
'Q
¯° p
¿
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
(21)
Lecture
Page 7 of 15
and this frequency averaged transmission represents the object of many years of work in
atmospheric radiative transfer by many people. We will discuss this more completely when we
discuss the HITRAN & MODTRAN transmission and radiation models later.
Case III:
Let us go back to Eq. 9a and consider the source function under conditions when the solar and
diffuse radiation field is much greater than the Planck emission. The approximate separation of
solar radiation and planetary emission will again be invoked to examine the radiative transfer
problem in the visible portion of the spectrum where FQ T !! B Q, WQ and the scattered radiation,
I T ', I, WQ !! B Q, WQ As before, the formal solution is:
W
I W, P, P0 , I, I0 n
³ J W ', P, P
0
, I, I0 e
WW ' P
dW '
0
W0
I W, P, P0 , I, I0 p
³ J W ', P, P0 , I, I0 e
W ' W P
W
P!0
P
dW '
P
P0
(22)
where J W ', P, P0 , I, I0 Q
Z
4
³ ³ P P, I, P ', I ' I W ', P, I, P ', I ' dP ' dI '
ZQ
FQ
W
exp §¨ Q ·¸ P P, I, P0 , I0 P0 ¹
4
©
The solution to this problem requires knowledge of the distribution of scatterers, the optical
properties of the scatterers and their Phase Function (the probability that a photon incident
from a particular direction will be scattered into another specific direction). In general, we must
also deal with the complex problem of multiple scattering. We’ll investigate the process of Mie
Scattering and Absorption by spherical particles, having specified sizes and optical properties.
We’ll use a computer program that provides exact solutions for Mie Scattering and Absorption.
Then, we’ll be dealing with a computer model capable of computing the radiation field for
multiple scattering, using a procedure known as Discrete Ordinates which divides the radiation
field into Fourier components and integrates the set of independent equations using Gaussian
quadrature.
2. Radiation Intensity and Radiation Flux
a. Radiation Intensity – amount of energy per unit time contained in an element of solid
angle which flows through a cross section of unit area perpendicular to the direction of
the beam.
Let us consider that we have isotropic radiation of intensity Io falling on one face of a
horizontal slab:
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 8 of 15
Solid Angle (in spherical coordinates)
b. Radiation Flux in Vertical Direction – amount of energy per unit time crossing a unit
surface perpendicular to the z direction. Let us integrate over the upper hemisphere.
2S S 2
Fp
³ ³
0
2S S 2
I sin T dT dI
0
³ ³I
o
0
sin T cos T dT dI
0
S2
2SI0
³ cos T sin T dT
S I0
0
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 9 of 15
For Isotropic Radiation, the flux is S times the intensity of a straight beam.
Applications using Radiation Intensity
1. Remote sounding
2. Satellite measurements
3. Target detection over horizontal/vertical paths
Applications using Flux:
1. Heating/Cooling of atmosphere
2. Radiation effects on climate.
3. Approximate Solution for Planetary Radiation (See Goody & Yung, pp. 57-63).
From Eq. 12, we have:
P
dIQ
dW Q
(23)
IQ BQ
This can be transformed into an integral equation by integrating both sides over all angles:
2S 1
dIQ
³ ³ dW
0 1
d
dW Q
2S 1
P dP dI
Q
³ ³I
Q
dP dI 4SBQ
0 1
§ 1
·
¨¨ 2S ³ P IQ dP ¸¸
© 1
¹
divergence of net
upward flux
ªIf IQ I0
« 1
«
I dP
« ³ Q
¬ 1
1
2S ³ IQ dP 4SBQ
1
total flux
4S IQ cons tan t º
»
»
2IQ
»
¼
total flux in
an LTE enclosure
ª d
º
SFQ »
«
d
W
¬ Q
¼
leading to the radiative transfer equation in net-flux form:
1 dFQ
4 dWQ
(24)
IQ BQ
Now – multiply both sides by P and integrate over all angles:
0
2S 1
³ ³P
2
dIQ
2 S 1
d P dI
0 0
dW v
d
dW Q
ª 1 2
º
«2S ³ P IQ dP »
¬ 1
¼
2S 1
³ ³ PI
Q
dP d I B Q
0 1
³ ³ P dP dI
0 1
SFQ
net upward flux
S KQ
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 10 of 15
dK Q
dW Q
(25)
FQ
Now, IQ looks something like this
In order to solve these equations, Eddington proposed a two-stream approximation:
IQ P IQ
0dP d1
IQ
0 ! P t 1
Thus
2S 1
³ ³I
4S IQ
Q
dP dI 2S IQ IQ
(26a)
0 1
2S 1
SFQ
³ ³ PI
Q
dP dI S IQ IQ
(26b)
0 1
2S 1
SK Q
³ ³P
0 1
1 dFQ
4 dWQ
2
IQ dP dI 2 S IQ IQ
3
1
2
I
Q
IQ BQ
(26c)
(26d)
where we have used Eq. 24.
2 d IQ IQ
3 dW Q
FQ
(27)
where we have used Eq. 26c.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 11 of 15
Or, differentiate Eq. 26d and use Eq. 27:
1 d2FQ
4 d W Q2
dBQ
1 d IQ IQ 2 dW Q
dW Q
1 d2FQ
4 d W Q2
dBQ
3
FQ 4
dW Q
d2FQ
3FQ
d W Q2
4
dBQ
dW Q
(28)
This is known as Eddington’s Equation.
The two required boundary conditions are usually given in the form of I- or I+. We have from Eq.
26d.
1 IQ
2
1 dFQ
1
BQ IQ
4 dWQ
2
Using 26d
1 dFQ
1 BQ IQ FQ
4 dWQ
2
where we have made use of Eq. 26b.
Thus: IQ
1 dFQ
1
BQ FQ
4 dW Q
2
EsQ BsQ (at bottom eg cloud-top or surface)
FQ at top sin ce IQ
and IQ
0 at WQ
0
IQ FQ
1
1 dFQ
BQ FQ
4 dW Q
2
(From I+ equation above)
= 0 (at top since no downward diffuse radiation)
To provide some simple analytical solutions it is useful to consider the grey approximation
wherein DQ is replaced by D (= grey absorption coefficient).
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 12 of 15
Thus, equation of radiative transfer can be integrated over frequency since
dWQ D Q dz Ÿ dW D dz
which is now independent of Q. Using the notation
P
dI
dW
³
4
dB
dW
1 dF
1
B F
4 dW
2
EsBs at bottom or F at top
1 dF
1
B F
4 dW
2
0 at top . Also note that B
I
dQ
IJ
d2 F
3F
dW2
I
ST 4
from earlier lecture.
S
where s=Stefan’s constant.
Example: Suppose we have an atmosphere at rest (i.e. no dynamical or latent heat fluxes).
Suppose also that net radiative heating is zero everywhere – that is the net upward flux S F =
constant (i.e. non-divergent). This state is called radiative equilibrium. Eddington’s equation is
now:
F
I
I
4 dB
3 dW
B
1
F
2
F at top
B
1
F
2
0 at top or B
1
F at top
2
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 13 of 15
i.e. B 0 E W
³
dB
E 0
1
F
2
W
3
F
4
³ dW
0
3
FW
4
B W B 0
B W 1
F
2
3
FW
4
B W
1·
§
F ¨3 W ¸
4
2¹
©
At the top of the atmosphere we need for the planetary average:
net incoming solar flux = net outgoing planetary flux
1 A SSa2
4Sa2
F
1 A B W
s T4
S
SF
S
4S
1 A S §3
1·
W ¸
¨
2¹
4S
©4
1 A S §3
1·
W ¸
¨
2¹
4S
©4
T4
ª 1 A S º § 3
1·
«
» ¨ W ¸
4s
2¹
«¬
»¼ © 4
Where s = Stetan-Boltzmann constant.
This is the simplest expression of the “greenhouse effect”.
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 14 of 15
1 A S
is written as Te4 where Te is called the “effective temperature”
4s
of the planet. (Te = 254.1K for Earth).
Notes: 1. Sometimes
2. In radiative equilibrium, the temperature of the surface is not equal to the temperature of
the air immediately above the surface. In particular at z=0 (or W=Ws):
I
B Ws 1
F
2
from previous page where B W EsBs Es Bs
B Ws Es Bs
1
F
2
1·
§
F ¨3 W ¸
4
2¹
©
1· 1
§3
F ¨ Ws ¸ F
2¹ 2
©4
§3
·
F ¨ Ws 1 ¸
4
©
¹
Ts 4
ª 1 A S º
«
» 3 Ws 1
«¬ 4sEs »¼ 4
surface temp. in
radiative
equilibrium
for the surface temperature in radiative equilibrium.
For the earth, let us take:
Es 1
A = 0.3
S = 1.35 x 106 erg cm-2 sec-1
s = 5.67 x 10-5 erg cm-2 deg-4 sec-1
h
W Ws exp z
Ws 4
h = scale ht. of principal
atmospheric absorber
(H2O) 2 km.
And we obtain:
Ts = 359.3K
What is temp. of atmosphere near surface?
What is temperature gradient of atmosphere at surface?
What is temperature at top of atmosphere?
12.815, Atmospheric Radiation
Dr. Robert A. McClatchey and Prof. Sara Seager
Lecture
Page 15 of 15