MET 61
MET 61 Introduction to Meteorology - Lecture 8
“Radiative Transfer”
Dr. Eugene Cordero
San Jose State University
Class Outline:
• Absorption and emission
• Scattering and reflected light
• Global Energy Balance
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MET 61 Introduction to Meteorology
Radiation Emission
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• B - Monochromatic Irradiance (Plank’s Law)
• F - Irradiance (Stefan Boltzmann Law)
• max – Peak emission at a wavelength
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MET 61 Introduction to Meteorology
Energy distribution
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• Radiative energy propagates at speed of light.
• Energy per unit area decrease as square of distance
from emitter:
 R1 

F2  F1 
 R2 
2
R1,, R2=radius
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MET 61 Introduction to Meteorology
Example
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• Estimate the value of the solar constant; the
irradiance at the top of the Earth’s atmosphere.
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MET 61 Introduction to Meteorology
Solution
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earth
sun
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MET 61 Introduction to Meteorology
Example
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• Estimate the value of the solar constant; the
irradiance at the top of the Earth’s atmosphere.
Given

5780
K
Given:::T
Tsun

5780
K
Given
T

5780
K
sun
sun
Given : Tsun  55780 K
55 km
R

6
.
9
x
10
sun
R
sun  6.9 x10 5 km
Rsun
sun  6.9 x10 km
88
8 km
R

111...49
xxx10
Rearth

49
10
km
R

49
10
8 km
earth
earth
Rearth
 14.49 x10 km 8
*
444
44
88

111 
444
777

222
*



E

67
10
Wm
K
5780
K
32
10
Wm
F
4  666...32

F111* 

TT
555...67
xxx10
K
xxx10
E


T

67
10
Wm
K
5780
K
32
10
Wm
4
8Wm
1K
 45780
7Wm
2
E  T  5.67 x10 Wm K 5780 K   6.32 x10 Wm
1
*
*
22*
2
2
E
F
F
E
E

222
111 2
1
222
2
 RR

R

E
F  R 

E

 E RR
R

 R 
*
*
111*
1

222
2
S-Solar Constant
555


6
.
96
x
10
*
7
22  6.96 x10 5 km
22
7



km
*
7

2

2


F
F22* 
66..32
xx10
Wm

1372
Wm


6
.
96
x
10
km


E

32
10
Wm

1372
Wm
7

2

2
8
2
8


1
.
45
x
10
km
E2  6.32 x10 Wm  1.45 x108 km   1372Wm
 161

.45
x10 km
MET
Introduction
to Meteorology
8
Absorption, Reflection and
Transmission
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• - emissivity: Fraction of blackbody that is actually
emitted (0-1)
• a - absorptivity: fraction of radiation striking an object
that is absorbed.
• t - transmissivity: fraction of radiation striking an
object that is transmitted.
• r - reflectivity: fraction of radiation striking an object
that is reflected.
• Energy is conserved, so:
a + r + t = 1
MET 61 Introduction to Meteorology
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Or in terms of irradiance
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B absorbed
a 
 absorptivity
B incident
B transmitted
t 
 transmissi vity
B incident
B reflected
r 
 reflectivity
B incident
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MET 61 Introduction to Meteorology
Kirchhoff’s law
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• Describes how good emitters are also good absorbers
a  
• This relationship is wavelength dependent.
• Albedo considers the net effect over a range of
wavelengths.
Freflected
A
Fincoming
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MET 61 Introduction to Meteorology
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Activity 7 Inclass question:
If the Earth’s albedo was to increase by 10%:
• A) By how much would surface solar radiation
change?
• B) How would the Earth’s surface energy budget
change?
• C) How would the Earth’s top of the atmosphere
budget change?
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MET 61 Introduction to Meteorology
Energy Balance
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• Energy at any level must be in balance:
Energy in = Energy out
Example:
Calculate the blackbody temperature of the
earth assuming a planetary albedo of 0.3 and
that the earth is in radiative equilibrium
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MET 61 Introduction to Meteorology
Solution
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• F (in; solar) = F (out; terrestrial)
S
F
S(1 - A)πR  E(4πR )
S(1
R ))
S(1 -- A)π
A)πR
R 
 E(4π
F(4πR
2
E  (1  A)S/4  241 W/m22
F

241
W/m
F
 (1
(1 
 A)S/4
A)S/4

241
W/m

TE  255  K
T
255
K
TEE 
MET
255
K to Meteorology
61 Introduction
2
2E
2
E
E
2
2
2E
E
E
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Example
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• A completely gray surface on the moon with
an absorptivity of 0.9 is exposed to overhead
solar radiation. What is the radiative
equilibrium temperature of the surface?
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MET 61 Introduction to Meteorology
Solution
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• Since the moon has no atmosphere, the
incoming solar radiation is the total incident
radiation upon the surface. For radiative
equilibrium:
absorbed ))  E
F
F((emitted
EE((absorbed
E
emitted))
2
4
2
4
2
4
a
x
1380
W/m



T
1368 W/m  TEEE
a x 1380
aa 
395KK
TTEE 395
E
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MET 61 Introduction to Meteorology
Atmospheric absorption
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• The amount of radiation that is absorbed by
the atmosphere is proportional to the number
of molecules per unit area that are absorbing.
B  B exp(   )

   sec   k dz
z
•
•
•
•
 (sigma) – optical depth or optical thickness
k- absorption coefficient (m2/kg)
 - density (kg/m3)
Angle of incidence (from vertical)
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MET 61 Introduction to Meteorology
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• So the transmissivity of the layer is now:
B transmitted
t 
 exp(   )
B incident
• And neglecting scattering, then the absorptivity is:
a   1  t  1  exp(   )
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MET 61 Introduction to Meteorology
Example
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• Parallel radiation is passing through a layer
100m in thickness containing a gas with an
average density of 0.1 kg/m3. The beam is
directed at 60° from normal to the layer.
Calculate the optical thickness and
transmissivity and absorptivity of the layer at
wavelength  where the absorption coefficient
is 10-1.
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MET 61 Introduction to Meteorology
Solution
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• Assuming the absorption coefficient and
density do not vary within the layer:
σ  k λλsecφ  ρdz
σ  10 11  2  0.1kg/m 33 *100m  2
• t=0.135
• a=0.865
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Sun angle
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MET 61 Introduction to Meteorology
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What month do you think this graph represents?
a) December b) March c) June d) September
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MET 61 Introduction to Meteorology
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What month do you think this graph represents?
a) December b) March c) June d) September
Answer: December
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MET 61 Introduction to Meteorology
Simplified radiative energy cascade for the
Earth-atmosphere climate system
Reflected
Planetary Albedo Extraterrestrial
Short Wave
Radiation
Energy
Input
Extraterrestrial
Short Wave
Radiation
Solar Temperature
E-A
Climate
System
Energy
Output
Terrestrial
Long Wave
Radiation
Planetary Temperature
MET 61
Assigned Reading for Feb 14
• Ahrens Ch 2 (continuing)
• Stull Ch 2: Pages 26-28
• Quiz 1 (30 minutes) on Feb
16th from material through
Feb 14th.
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MET 61 Introduction to Meteorology
Activity 7: Due March 21st
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• Question 1: Concrete has an albedo of around .25 and yet the
typical infrared emissivity of concrete is 0.8. Explain why these
are different and the implication of this on climate change?
• Question 2: Consider a flat surface subject to overhead
radiation. If the absorptivity is 0.1 for solar radiation and 0.8 in
the infrared, compute the radiative equilibrium temperature.
• Question 3: Calculate the radiative equilibrium temperature of
the Earth’s surface and Earth’s atmosphere assuming that the
earth’s atmosphere can be regarded as a thin layer with an
absorptivity of 0.1 for solar radiation and 0.8 for terrestrial
radiation. Assume the earth’s surface radiates as a blackbody
at all wavelengths.
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MET 61 Introduction to Meteorology