Transcript Presentation Slides for Chapter 9 of Fundamentals of Atmospheric Modeling 2nd Edition Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 [email protected] March 21, 2005

Presentation Slides
for
Chapter 9
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
[email protected]
March 21, 2005
Earth-Atmosphere Energy Balance
Fig. 9.1
Radiant energy per year --->
Radiant energy per year
Energy Transfer From Equator to Poles
Absorbed incoming solar
Surplus
Emitted
outgoing infrared
Deficit
Deficit
Transfer
-90
-60
-30
0
30
Latitude
60
90
Fig. 9.2
Electromagnetic Spectrum
Radiation in the form of an electromagnetic wave
Wavelength
(9.1)
c 1
 
˜
 
Radiation in the form of a photon of energy
Energy per unit photon (J photon-1)
hc
E p  h 

(9.3)
Electromagnetic Spectrum
-->
-->
-->
 = 0.5 m
Ep = 3.97 x 10-19 J photon-1
 = 5.996 x 1014 s-1
˜ = 2 m-1
-->
-->
-->
 = 10 m
Ep = 1.98 x 10-20 J photon-1
 = 2.998 x 1013 s-1
˜ = 0.1 m-1
Example 9.1:
Planck’s Law
Radiance
Intensity of emission per incremental solid angle
Planck radiance (W m-2 m-1 sr-1)
(9.4)
2hc 2
B,T 
 hc  
5 
 exp 
 1
 k B T  
Radiance actually emitted by a substance
e     B,T
(9.5)
Kirchoff’s law
In thermodynamic equilibrium, absorptivity (a) = emissivity
() --> the efficiency at which a substance absorbs equals that
at which it emits. --> a perfect emitter is a perfect absorber
Emissivities
Infrared emissivities of different surfaces
Surface Type
Liquid water
Fresh snow
Old snow
Liq. water clouds
Cirrus clouds
Ice
Emissivity
(fraction)
1.0
0.99
0.82
0.25 - 1.0
0.1 - 0.9
0.96
Surface Type
Soil
Grass
Desert
Forest
Concrete
Urban
Emissivity
(fraction)
0.9 - 0.98
0.9 - 0.95
0.84 - 0.91
0.95 - 0.97
0.71-0.9
0.85 - 0.87
Table 9.1
Solid Angle
Radiance emitted from point (O) passes through incremental area
dAs at distance rs from the point.
Incremental surface area
(9.7)
2
dAs  rsdrs sind  rs sindd
Incremental solid angle (sr) (9.6)
Steradians analogous to radians
dAs
da  2  sindd
rs
Solid angle around a sphere (9.9)
a 
2 
a da  0 0 sindd  4
Fig. 9.3
Spectral Actinic Flux
Integral of spectral radiance over all solid angles of a sphere
Used to calculate photolysis rate coefficients
Incremental spectral actinic flux
(9.10)
dE  Ida
Spectral actinic flux
E 
(9.11)
2 
a dE   a I da  0 0 I  sindd
Isotropic spectral actinic flux
2 
 0 s indd  4I 
E  I 
0
(9.12)
Spectral Irradiance
Flux of radiant energy propagating across a flat surface
Used to calculate heating rates
Incremental spectral irradiance
(9.13)
dF = Icosda
Integral of dF over the hemisphere above the x-y plane (9.14)
F 
2  2
 a dF  a I  cosd a  0 0
I  cossindd
Isotropic spectral irradiance
2  2
(9.15)
 
F  I
cos s indd  I
0 0
Spectral irradiance at the surface of a blackbody
F = I  B,T
(9.16)
Irradiance (W m-2 m-1)
Spectral Irradiance v. Temperature
Fig. 9.4
Emission Spectra of the Sun and Earth
104
Irradiance (W
m-2
-2
m-1)m
-1
)
Irradiance emission versus wavelength for the Sun and Earth
when both are considered blackbodies
Sun
102
Visible
0
10
-2
Ultraviolet
10
Infrared
Earth
-4
10
0.01
0.1
1
10
Wavelength m)
(
100
Fig. 9.5
Ultraviolet and Visible Solar Spectrum
-2
4
1.2 10
1 104
Near
UV
4 103
2 103
0.1
0.2
Red
3
6 10
Green
Blue
8 103
Far
UV
UV-A
UV-B
-1
m

Irradiance (W m-2 m-1) )
Ultraviolet and visible portions of the solar spectrum.
Visible
0.3 0.4 0.5 0.6
Wavelength m)
(
0.7
0.8
Fig. 9.6
Wien’s Displacement Law
Differentiate Planck's law with respect to wavelength at constant
temperature and set result to zero
Peak wavelength of emissions from blackbody
(9.17)
2897
 p m 
TK
Example 9.2:
Sun’s photosphere
p = 2897/5800 K
= 0.5 m
Earth’s surface
p = 2897/288 K
= 10.1 m
Wien’s Displacement Law
0.5 m
4
10
-2
-1
m
)
Irradiance (W m-2 m-1
)
Gives line through peak irradiances at different temperatures.
102
100
6000 K
4000 K
2000 K
1000 K
-2
10
-4
10
0.01
10 m
300 K
0.1
1
10
Wavelength m)
(
100
Fig. 9.7
Stefan-Boltzmann Law
Integrate Planck irradiance over all wavelengths
Stefan-Boltzmann law (W m-2)
Fb  
(9.18)

0
B,T d   B T 4
Stefan-Boltzmann constant
2kB4 4
8
-2 K-4
W
m
B 

5.67
10
3 2
15h c
Example 9.3:
T = 5800 K
---> FT = 64 million W m-2
T = 288 K
---> FT = 390 W m-2
Reflection and Refraction
Reflection
Angle of reflection equals angle of incidence
Refraction
Angle of wave propagation relative to surface normal changes as
the wave passes from a medium of one density to that of another
Fig. 9.8
Reflection
Albedo = fraction of incident sunlight reflected
Albedos in the non-UVB solar spectrum
Surface Type
Albedo (fraction)
Earth & atmosphere
0.3
Liquid water
0.05 - 0.2
Fresh snow
0.75 - 0.95
Old snow
0.4 - 0.7
Thick clouds
0.3 - 0.9
Thin clouds
0.2 - 0.7
Sea Ice
0.25 - 0.4
Surface Type
Soil
Grass
Desert
Forest
Asphalt
Concrete
Urban
Albedo (fraction)
0.05 - 0.2
0.16 - 0.26
0.20 - 0.40
0.10 - 0.25
0.05 - 0.2
0.1 - 0.35
0.1 - 0.27
Table 9.2
Refraction
Snell’s law
(9.19)
n2 sin1

n1 sin 2
Real part of the index of refraction (≥1)
(9.20)
Ratio of speed of light in a vacuum to that in a given medium
c
n1 
c1
Real part of the index of refraction of air
8 
(9.21)
15, 997 
na, 1  10 8342.13


2
2

130 
38.9   
2, 406,030
Real Refractive Indices v Wavelength
Wavelength (m)
0.3
0.5
1.0
10.0
Air
Water
1.000292
1.000279
1.000274
1.000273
1.349
1.335
1.327
1.218
Table 9.2
Refraction
--->
--->
--->
--->

1
nair
nwater
2
cair
= 0.5 m
= 45o
= 1.000279
= 1.335
= 32o
= 2.9971 x 108 m s-1
--->
cwater
= 2.2456 x 108 m s-1
Example 9.4:
n2 sin1

n1 sin 2
Total Internal Reflection
Critical angle
(9.22)
1 n1
o 
 2,c  s in  s in90 
 n2

Example 9.5:
--->
--->
--->

nair
nwater
2,c
= 0.5 m
= 1.000279
= 1.335
= 48.53o
Geometry of a Primary Rainbow
Fig. 9.9
Diffraction Around A Particle
Huygens' principle
Each point of an advancing wavefront may be considered the
source of a new series of secondary waves
Fig. 9.10
Radiation Scattering by a Sphere
Ray A is reflected
Ray B is refracted twice
Ray C is diffracted
Ray D is refracted, reflected twice, then refracted
Ray E is refracted, reflected once, and refracted
Fig. 9.11
Forward and Backscattering
Cloud droplets
Scatter primarily in the forward direction
Gas molecules
Scatter evenly in the forward and backward directions.
Fig. 9.12
Change in Color of Sun During the Day
Fig. 9.13
Gas Absorption
Gas
Absorption wavelengths (m)
Visible/Near-UV/Far-UV absorbers
Ozone
< 0.35, 0.45-0.75
Nitrate radical
< 0.67
Nitrogen dioxide
< 0.71
Near-UV/Far-UV absorbers
Formaldehyde
< 0.36
Nitric acid
< 0.33
Far-UV absorbers
Molecular oxygen
Carbon dioxide
Water vapor
Molecular nitrogen
< 0.245
< 0.21
< 0.21
< 0.1
Table 9.4
Fraction of transmitted radiation through all important gases
Fraction transmitted
Fraction transmitted
Gas Absorption
1
Line-by-line
Model
0.8
0.6
Eleven gases
0.4
0.2
0
1
10
100
Wavelength m)
(
1000
Fig 9.14
Fraction of transmitted radiation through water vapor
1
Fraction transmitted
Fraction transmitted
Gas Absorption
H2 O
0.8
Line-by-line
Model
0.6
0.4
0.2
0
1
10
100
Wavelength m)
(
1000
Fig 9.14
Gas Absorption
Fraction transmitted
1
0.8
CO2
0.6
0.4
Line-by-line
Model
0.2
0
1
10
100
Wavelength m)
(
1000
Fig 9.14
Gas Absorption
Fraction transmitted
1
0.8
O3
0.6
0.4
Line-by-line
Model
0.2
0
1
10
100
Wavelength m)
(
1000
Fig 9.14
Fraction transmitted
Gas Absorption
1
0.95
0.9
0.85
CH4
Line-by-line
Model
0.8
0.75
0.7
1
10
100
Wavelength m)
(
1000
Fig 9.14
Extinction Coefficient
Attenuation of incident radiance, Io, due to absorption as it travels
through a column of gas.
Extinction coefficient () (cm-1, m-1, or km-1)
A measure of the loss of radiation per unit distance
Fig 9.15
Extinction Coefficient
Reduction in radiance with distance through a gas
(9.23)
dI 
 Nq ba,g,q,,T I   a,g,q,,T I
dx
Integrate
(9.24)
I  I0, eNq ba, g,q, , T xx0   I 0,ea, g,q, , T xx 0 
Extinction coefficient due gas absorption
 a,g, 
Transmission
Na g
Na g
q 1
q 1
(9.25)
 Nq ba,g,q,,T    a,g,q,,T
(9.29)
I
Ta,g,q, 
 e uq ka, g,q,   e  a, g,q, xx 0 
I0
Absorption Coefficient
Extinction coefficient in terms of mass absorption
(9.26)
Na g
Na g
Nag
N q mq
uq
 a,g, 
 q k a,g,q, 
ka,g,q, 
k a,g,q,
A
x  x 0 

2
2 /g)/g)
Absorption coefficient (cm
Absorption coefficient (cm
q 1


q1
q1
2
10
1
322.15 hPa
22.57 hPa
10
0
10
-1
10
-2
10
10-3
6.619 6.6205 6.622 6.6235 6.625
Wavelength m)
(
Fig 9.16
Absorption Coefficient
Absorption Coefficient
(9.27)


Sq T q pa , T

A 1 
k a,g,q, 


2
m q  
˜   ˜q   q pref pa 
 q pa , T 2  


 
Pressure-broadened half-width
  
(9.28)
nq
T
 re f 
 q pa ,T  
  air,q pre f , Tre f pa  pq   se lf ,q pre f , Tre f pq
 T 





 
Transmission Example
Monochromatic transmission
(9.29)
 
I
Ta,g,q, uq 
 e uq ka, g,q,   e  a,g, q, xx0 
I0
Exact transmission when two absorption lines

T
˜ ,
˜ ˜u  0.5 e
k x u
e
k y u
(9.30)

Transmission overestimated when lines averaged
T
˜ ,
˜ ˜u  e


0.5 k x ky u
(9.31)
Correlated k-Distribution Method
Exact transmission in wavenumber interval
(9.32)
  k u
1 ˜ ˜ k ˜u
˜  e
T
e
d
f
˜ ,
˜ ˜u 
˜ k dk 

˜
˜ 
0

Integration of differential probability is unity
(9.33)

0 f˜k dk   1
Reorder absorption coefs. into cumulative frequency distribution
(9.34)
g ˜k  
k
0 f˜ k dk 
Effects on Visibility of Gas Absorption
Meteorological range (Koschmieder equation)
3.912
x
 ext,
Meteorological ranges due to Rayleigh scattering and NO2 absorption
Wavelength
(mm)
0.42
0.50
0.55
0.65
Rayleigh Scat.
(km)
112
227
334
664
<-- NO2 absorption -->
0.01 ppmv
0.25 ppmv
(km)
(km)
296
11.8
641
25.6
1,590
63.6
13,000
520
Table 9.5
Effects on Visibility of Gas Absorption
-1
Extinction coefficient (km-1))
Extinction coefficient due to NO2 and O3 absorption.
101
100
10-1
-2
10
10-3
-4
10
-5
10
10-6
NO (g) (0.25 ppmv)
2
NO (g) (0.01 ppmv)
2
O (g) (0.25 ppmv)
3
O (g) (0.01 ppmv)
3
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Wavelengthm)
(
Fig. 9.17
Gas Scattering
Rayleigh scatterer: 2r/ <<1
Extinction coefficient due to Rayleigh scattering
(9.35)
 s,g,  Na bs,g,
Scattering cross section of a typical air molecule (cm2) (9.36)


2
2
2
3
8 na,  1
32 na,  1
bs,g, 
f *  
f * 
4 2
4 2
3 Na,0
3 Na,0
3
Anisotropic correction factor
6  3 *
f *  
 1.05
6  7*
(9.37)
Rayleigh Scattering Example
Example 9.6:
--->
--->

pa
T
s,g,
x
= 0.5 m
= 1 atm (sea level)
= 288 K
= 1.72 x 10-7 cm-1
= 227 km
--->
--->

= 0.55 m
s,g, = 1.17 x 10-7 cm-1
x
= 334 km
Imaginary Index of Refraction
Measure of extent to which a substance absorbs radiation
Attenuation of
incident radiance, I0,
due to absorption
Equation for
attenuation
dI
4

I
dx

Integrate
I  I0e
Fig 9.18
(9.38)
(9.39)
4 xx0  
Complex Index of Refraction
m  n i
(9.40)
Real and imaginary refractive indices at = 0.5 and 10 m
Substance
Liquid water
Black carbon
Organic matter
Sulfuric acid
<-- 0.5 m -->
Real Imaginary
1.34 1x10-9
1.82 0.74
1.45 0.001
1.43 1x10-8
<-- 10 m -->
Real Imaginary
1.22 0.05
2.4
1.0
1.77 0.12
1.89 0.46
Table 9.6
Transmission
Light transmission through particles at  = 0.5 m
Diameter (m)
<-- Transmission (I/I0) -->
Black carbon
Water
(=0.74)
(=1x10-9)
0.1
1.0
10
0.16
8x10-9
0
0.999999997
0.99999997
0.9999997
Table 9.6
Imaginary index of refraction
Imaginary Refractive Index of Liquid
Nitrobenzene
0.5
0.4
0.3
0.2
0.1
0
0.2
0.3
0.4
0.5
0.6
Wavelength m)
(
0.7
Fig. 9.19
Particle Extinction Coefficients
Particle absorption/scattering extinction coefficients
 a,a, 
(9.41)
NB
 niba,a,i,
i1
 s,a, 
NB
 nibs,a,i,
i1
Particle absorption/scattering cross sections
2
ba,a,i,  ri Qa,i,
bs,a,i,  ri2Qs,i,
(9.42)
Tyndall Absorption / Scattering
Rayleigh regime (di<0.03 or ai,<0.1)
Size parameter
a i,  2ri 
(9.43)
Tyndall absorption efficiency (linear with ri/)
(9.44)
2 

 2
2ri  m  1  2ri 
24n 
Qa,i,  4
Im

2



  2
2 2
2
2
 m   2 

n



4
n






 1
 

 

(9.45)


2ri  24n   
  0 --> Qa,i, 
--> linear with 


2

2

n
   2 








Tyndall Absorption / Scattering
Tyndall scattering efficiency [linear with (ri/]
8  2ri 4

Qs,i,  
3   
Example 9.7 (Liquid water):
--->
--->
--->
--->

ri


Qs,i,
Qa,i,
= 0.5 m
= 0.01 m
= 1.34
= 1.0 x 10-9
= 2.9 x 10-5
= 2.8 x 10-10
2
2
m 1

2
m  2
(9.46)
Mie Absorption / Scattering
Mie regime (0.03< di <32 or 0.1< ai,<100)
Single particle Mie scattering efficiency
2
Qs,i, 
a i,

2k  1ak
(9.47)
2
 bk
k1
2

Single particle Mie absorption efficiency
Qa,i,  Qe,i,  Qs,i,
Single particle total extinction coefficient
Qe,i, 
2
a i,

 2k  1Reak  bk 
k1
(9.48)
Soot Absorption/Scattering Efficiencies
Single Particle Absorption/Scattering Efficiency at  = 0.50 m
1.5
1
0.5
2
Mie regime
Rayleigh regime
Rayleigh regime
2
Geometric
regime
Q
1.5
s
1
Qa
Q
0.5
f
0
0.01
0.1
1
10
100
P article diameterm)
(
0
1000
Fig. 9.20
Water Absorption/Scattering Efficiencies
Rayleigh regime
Single Particle Absorption/Scattering Efficiency at  = 0.50 m.
5
3
2
1
0
0.01
Rayleigh regime
qsc
4
Mie regime
Geometric
regime
103
101
10-1
10-3
-5
10
Q
f
Q
s
0.1
Q
a
1
10
100
P article diameterm)
(
-7
10
-9
10
1000
Fig. 9.21
Geometric Absorption / Scattering
Geometric regime (di >32 or ai,>100) --> significant diffraction
In the limit (ai,-->∞), scattering efficiency is constant (9.49)
li m Qs,i,
a i,  
 1
m 1
m  2
Also, as ai,-->∞, Qs,i,≈Qs,i, regardless of how weak the
imaginary index of refraction is.
Example 9.8:

--->
--->
--->

Qs,i,
Qa,i,
= 0.5 m
= 1.34 for liquid water
= 1.1 as ai,-->∞ from9.49
= 1.1 as ai,-->∞ from Fig. 9.21
Mixing Rules
Volume average
(9.50)
m 
NV 
q

q1

m ,q 
NV 
q
  n  i 
q1
Volume average dielectric constant
m 2 
NV 
q

q1

m 2,q 
NV 
q

q1

 ,q 
(9.54)
NV 
q
   r,  ii, 
q1
Complex dielectric constant
(9.51)
2
2
2
2
  m  n  i    n    i2n   r,  ii,
Mixing Rules
Real/imaginary complex dielectric constant
r,  n2   2
(9.52)
 i,  2 n  
Real and imaginary refractive indices
n 
2
2
 r,   i,   r,
2
(9.53)
2
 
2
 r,   i,   r,
2
Mixing Rules
Maxwell Garnett
(9.55)
NA 


A,q   ,A,q    ,M 


3




   ,A,q  2 ,M  

q1
2
m       ,M 1 

NA
A,q   ,A,q   ,M  



 1 



   , A,q  2 ,M  

q 1




Bruggeman
(9.56)
 N A 

A,q   ,M    



 1




   , A,q  2   
   ,M  2  
q 1
q1


NA 
A,q   ,A,q    


Absorption cross section
Enhancement factor
Absorption cross section enhancement due to internal mixing
enhancement factor
Absorption cross section
Absorption Cross Section Enhancement
6
5
4
d
BC
(m )
=0.5 m
0.04
0.08
0.12
3
2
1
0
0.01
0.1
1
10
100
Total particle (soot core+S(VI) shell) diameter
m)
(
Fig. 9.22
Modeled Extinction Coefficient Profile
0 0.1 0.2 0.3 0.4 0.5
Claremont
8/27/97
11:30
0.32 m
400
Pressure (hPa)
Pressure (hPa)
300
500
600
Gas scat.
700
Gas abs. Aer. abs.
Aer. scat.
800
900
1000
0
0.1 0.2 0.3 0.4 0.5
(km-1 )
Fig. 9.23a
Modeled Extinction Coefficient Profile
0 0.1 0.2 0.3 0.4 0.5
Claremont
8/27/97
11:30
0.61 m
400
Pressure (hPa)
Pressure (hPa)
300
500
600
Gas scat.
Gas abs.
700
800
Aer. abs.
Aer. scat.
900
1000
0
0.1 0.2 0.3 0.4 0.5
(km-1 )
Fig. 9.23b
Visibility Definitions
Meteorological range
Distance from an ideal dark object at which the object has a
0.02 liminal contrast ratio against a white background
Liminal contrast ratio
Lowest visually perceptible brightness contrast a person can see
Visual range
Actual distance at which a person can discern an ideal dark
object against the horizon sky
Prevailing visibility
Greatest visual range a person can see along 50 percent or more
of the horizon circle (360o), but not necessarily in continuous
sectors around the circle.
Visibility Definitions
The intensity of radiation increases from 0 at point x0 to I at point x
due to the scattering of background light into the viewer’s path
Fig. 9.24
Meteorological Range
Contrast ratio
IB  I
Cratio 
IB
(9.57)
Change in object intensity along path of radiation
dI
 t I B  I 
dx
Total extinction coefficient
 t   a,g  s,g  a,p  s,p
(9.58)
Integrate (9.58) and substitute into (9.57)
IB  I
 t x
Cratio 
e
IB
Meteorological range (Koschmieder equation)
(9.60)
3.912
x
t
(9.59)
(9.61)
Meteorological Range
Meteorological Range (km)
Gas
Gas
Particle
Particle
scattering absorption scattering absorption
All
Polluted
day
366
130
9.59
49.7
7.42
Lesspolluted
day
352
326
151
421
67.1
(Larson et al., 1984)
Table 9.8
Optical Depth
Total extinction coefficient
(9.62)
    s,g,   a,g,  s,a,  a,a,  s,h,   a,h,
Incremental distance versus incremental path length
(9.63)
dz  cossdSb  sdSb
Incremental optical depth
(9.66)
d  dz  sdSb
Optical depth as a function of altitude
 
z
Sb
   dz  
(9.63)
  s dSb
Optical Depth
Relationship between optical depth, altitude, solar zenith angle,
and pathlength
Fig. 9.22
Solar Zenith Angle
Cosine of solar zenith angle
(9.67)
cos s  sin sin   cos  cos  cos H a
Solar declination angle ()
Angle between the equator and the north or south latitude of
the subsolar point
Subsolar point
Point at which the sun is directly overhead
Local hour angle (Ha)
Angle, measured westward, between longitude of subsolar
point and longitude of location of interest.
Solar Zenith Angle
Geometry for zenith angle calculations.
Fig. 10.26 b
Solar Declination Angle
Solar declination angle
(9.68)
1
  sin
sinob sinec
Obliquity of the ecliptic
(9.69)
Angle between the plane of the Earth's equator and the plane of
the ecliptic, which is the mean plane of the Earth's orbit around
the Sun.
 ob  23o .439 0o .0000004N JD
Solar Declination Angle
Ecliptic longitude of the sun
(9.71)
ec  LM  1o.915sing M  0 o.020sin2gM
Mean longitude of the sun
(9.72)
LM  280o.460 0 o.9856474NJD
Mean anomaly of the sun
g M  357o.528 0o.9856003NJD
(9.72)
Solar Zenith Angle
Local hour angle
(9.73)
2ts
Ha 
86, 400
Example:
At noon, when sun is directly overhead, Ha = 0 --->
cos s  sin  sin   cos  cos 
When the sun is over the equator,  = 0 --->
coss  cos cosHa
Solar Zenith Angle
Example 9.9:
1:00 p.m., PST, Feb. 27, 1994,  = 35 oN
---> DJ = 58
---> NJD = -2134.5
---> gm = -1746.23o
---> Lm = -1823.40o
---> ce = -1821.87o
---> ob = 23.4399o
--->  = -8.52o
---> Ha = 15.0o
o
o
o
o
o
---> coss  sin 35 sin 8.52  cos 35 cos 8.52 cos15.0
  
---> s
= 45.8o
   
  
Solstices and Equinoxes
Solar declinations during solstices and equinoxes. The Earth-Sun
distance is greatest at the summer solstice.
Sun
Sun
N.H
.
sum
sols mer,
tice S.H
.
, Ju
ne 2 winter
2
Vernal equinox, Mar. 20
Tropic of
Cancer
Axis of
rotation
Equator
23.5o
Autumnal equinox, Sept. 23 23.5o
Sun
er
m
sum
.
Topic of
H
.
2
S
r,
.2
e
c
t
Capricorn
e
n
i
D
w
,
.
ce
N.H solsti
Earth
Fig. 9.27
Radiative Transfer Equation
Change in radiance / irradiance along a beam of interest
Change in radiance along incremental path length
(9.74)
dI   dIso,  dIao,  dIsi,  dI Si,  dI e i,
Scattering of radiation out of the beam
(9.75)
dI so,  I  s,dSb
Absorption of radiation along the beam
dI ao,  I  a,dSb
(9.76)
Radiative Transfer Equation
Multiple scattering of diffuse radiation into the beam
(9.77)

  s,k, 2 1

dI si,   k 
I, ,  Ps,k,,,,,
  dddSb


0
1
 4


Single scattering of direct solar radiation into beam
(9.78)

  s,k ,

dI Si,   k 
Ps,k,, s ,,s Fs, e    s dSb
 4


Emission of infrared Planckian radiation into beam
dI e i,  a, B,T dSb
(9.79)
Extinction Coefficients
Extinction due to total scattering
(9.80)
 s,   s,g,   s,a,   s,h,
Extinction due to total absorption
(9.80)
 a,   a,g,   a,a,   a,h,
Extinction due to total scattering plus absorption
    s,  a,
(9.81)
Scattering Phase Function
Gives angular distribution of scattered energy vs. direction
Scattering phase function for diffuse radiation
Ps,k , ,, ,,
   redirects diffuse radiation from ’, ’ to , 
Scattering phase function for direct radiation
Ps,k , ,, s ,, s redirects direct solar radiation from - ,  to ,

Scattering Phase Function
Single scattering of direct solar radiation and multiple scattering of
diffuse radiation.
Fig. 9.28
Scattering Phase Function
Scattering phase function defined such that
(9.82)
1
Ps,k,  da  1

4 4
 = angle between directions ’, ’ and , 
Substitute
da sindd -->
1 2 
Ps,k ,  sindd  1


4 0 0
(9.83)
Scattering Phase Function
Phase function for isotropic scattering
(9.84)
Ps,k ,    1
Phase function for Rayleigh scattering

(9.85)

3
Ps,k ,    1  cos2 
4
Henyey-Greenstein function
Ps,k,   

(9.86)
2
1  ga,k,

32
2
1 ga,k,  2ga,k, cos
Scattering Phase Function
Scattering phase functions for (a) isotropic and (b) Rayleigh
scattering
1.5
1
0.5
1.5
1
Ps ()

0
0.5
-0.5
-1
-1
-1.5
-1.5
1 0.5
0 -0.5 -1 -1.5 -2
(a)

0
-0.5
2 1.5
Ps ()
2 1.5
1 0.5
0 -0.5 -1 -1.5 -2
(b)
Fig. 9.29
Asymmetry Factor
First moment of phase function -- relative direction of scattering
 0

ga,k,  0
 0

(9.88)
forward (Mi e) scatteri ng
isotropic or Rayleigh scatteri ng
backward scatteri ng
(9.87)
1
ga,k, 
Ps,k,  cosd a

4 4
Expand with da sindd --->
(9.89)
1 2 
ga,k, 
Ps,k,  cos s indd
4 0 0
 
Isotropic Scattering ---> Ps,k ,    1 --->
(9.90)
1 2 
ga,k, 
cos s indd  0
4 0 0
 
Asymmetry Factor
Rayleigh scattering --->
(9.91)


1 2  3
2
ga,k, 
1

cos
 cos s indd  0


4 0 0 4
Mie scattering --->
(9.92)
ga,k,  Q f,i, Qs,i,
Backscatter ratio
ba,k , 
2 
0  2 Ps,k,  s indd
2 
0 0 Ps,k,  s indd
(9.93)
Energy Emitted by Sun to Earth
Summed irradiance (W m-2) emitted at Sun's photosphere
Fp 
Lp
4


T
B p
2
4R p
Lp = emissions from Sun's photosphere = 3.9 x 1026 W
Rp = radius from Sun center to photosphere = 6.96 x 108 m
Tp = temperature of photosphere = 5796 K
Example 9.10:
Tp = 5796 K
--->Fp ≈ 6.4 x 107 W m-2
(9.94)
Solar Constant
Mean summed irradiance at top of Earth's atmosphere (9.95)
2
2
 R p 
 Rp 
4
Fs  
F


T
 p  R  B p
R
 es 
 es 
Calculated Fs
Measured
≈ 1379 W m-2
Fs ≈ 1365 W m-2
Varies by +/- 1 W m-2 over each 11 year sunspot cycle
Solar Constant
Daily solar flux depends on Earth-Sun distance
(9.96)
2
 Res 
Fs  
Fs

 Res 
Empirical formula
(9.97)
2
 Res 
 R   1.00011 0.034221cos J  0.00128sin J
 es 
0.000719cos2 J  0.000077sin2J
J  2DJ DY
Incident Solar Flux
Example 9.11:
December 22
---> Fs = 1365 x 1.034 = 1411 W m-2
June 22
---> Fs = 1365 x 0.967 = 1321 W m-2
--> Irradiance varies by 90 W m-2 (6.6%) between Dec. and June
Cumulative solar irradiance as sum of spectral irradiances (9.98)
2
2
 Res 
 Res 
Fs   Fs,   
  Fs,   
 Fs
Res 
Res 








Seasons
Relationship between the Sun and Earth during the solstices and
equinoxes
Fig. 9.30
Equilibrium Earth Temperature
Energy (W) absorbed by the Earth-atmosphere system (9.99)
 2 

Ein  Fs 1 Ae,0 Re
Energy emitted by the Earth's surface
4
(9.100)
 
2
Eout   e,0  BTe 4Re
Equate --> temperature without greenhouse effect

Fs 1 Ae,0
Te  
4 e,0  B



14


(9.101)
Equilibrium Earth Temperature
Example 9.12:
Fs
--->
Ae,0
Te
= 1365 W m-2
= 0.3
= 254.8 K
Actual average surface temperature on earth ≈ 288 K -->
difference due to absorption by greenhouse gases
Radiative Transfer Equation
dI ,,
dSb
(9.102)
  s,k, 2 1

 I ,, s,   a,   
I , ,
  Ps,k , ,, ,,
 dd


0
1
4


k

   s
 Fs,e

  s,k,

  4 Ps,k, ,, s ,, s    a, B,T
k
Single scattering albedo
(9.103)
 s,g,   s,a,   s,h,
 s,
 s, 


s,g,  a,g,   s,a,   a,a,   s,h,   a,h,
Total extinction coefficient
    s,  a,
Optical depth
d  sdSb
Radiative Transfer Equation
Rewrite radiative transfer equation

where
dI,,
d
(9.104)
diffuse
direct
emis
 I,,  J,,  J,,  J,,
(9.105)
 s,k, 2 1

d
   0 1 I ,, Ps,k,,,,,
 d 


k
(9.106)
 s,k,

1
dire ct
   s
J ,, 
Fs, e
Ps,k ,,, s ,,s 


4
  

1
diffuse
J ,, 
4
k
emis


J,,  1 s, B,T
(9.107)
Beer’s Law
Consider only absorption in downward direction

dI,,
d a,
(9.108)
 I , ,
Solution
(9.109)
 

 
I,,  a,  I ,,  a,,t e

  a,  a,,t 
Schwartzchild’s Equation
Consider absorption and infrared emissions

dI,,
d a,
(9.111)
 I , ,  B,T
Solution
(9.112)
 

 
I,,  a,  I ,,  a,,t e

  a,  a,,t 
1 a,  
  a, a,   
 
B,T a, e
d a,


  a,,t 

Two-Stream Method
Divide phase function into upward (+) and downward component
(9.114)








 1  g
1  ga,k,
a,k,

I 
I  u pward
2
2

1 2 1
I, ,  Ps,k,,,,,
  d d  
0
1
4
 1  g
1  ga,k,
a,k,

I 
I  d own war

2
2
 
Substitute (9.114) into (9.105)
(9.115)
 s, 1 b I   s, b I 


2
1


1
s,k,
I,, Ps,k,,,,,
d  


  d 


0
1
4

  1 b I   b I
k  
 s,
 
s,  
Two-Stream Method
Integrated fraction of forward scattered energy
(9.116)
1 ga,
1  b 
2
Integrated fraction of backscattered energy
(9.116)
1 ga,
b 
2
Effective asymmetry parameter
 s,a, ga,a,   s,c, ga,h,
ga, 
 s,g,   s,a,   s,h,
(9.117)
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
300
300
600
g
700
a
s
No aerosols
With aerosols
800
500
600
1000
1000
1
a
s
No aerosols
With aerosols
800
900
0.2 0.4 0.6 0.8
g
700
900
0
Claremont
8/27/97
11:30
0.61 m
400
Pressure (hPa)
500
Pressure (hPa)
Claremont
8/27/97
11:30
0.32 m
400
Pressure (hPa)
Pressure (hPa)
Modeled Asymmetry Parameter
0
0.2 0.4 0.6 0.8
Fig. 9.28
1
Two-Stream Approximation
Upward radiance equation
(9.119)
dI 
s
1
 I   s 1 bI   s bI  
1 3ga1s Fse  s
d
4
Downward radiance equation
(9.120)
dI 
s
1
 I   s 1 bI   s bI  

1  3ga1s Fs e   s
d
4
Irradiances in terms of radiance for two-stream approximation
F  21 I 
F  21 I 
Two-Stream Approximation
Substitute irradiances and generalize for different phase function
approximations
Solar irradiance
(9.121)
dF 
  1F   2 F   3  s Fs e   s
d
dF 
  1F   2 F  1   3  s Fs e   s
d
Surface boundary condition
Aes Fs e  N L 1 2  s
F  N L 1 2  Ae F  NL 1 2  
 BT
(9.123)
sol ar
infrared
Two-Stream Approximation
Coefficients for two stream approximations using two techniques
Approximation
1
2
 s 1  ga 
21
3
Quadrature
1   s 1  ga  2
1
Eddington
7  s 4  3ga  1  s 4  3ga  2  3ga s

4
4
4
1  3ga 1 s
2
Infrared irradiance
(9.122)
dF 
  1F   2 F  21 s BT
d
dF 
  1F   2 F  21  s BT
Table 9.10
d
Delta Functions
Quadrature and Eddington solutions underpredict forward
scattering because expansion of phase function is too simple to
obtain the strong peak in scattering efficiency.
--> adjust terms with delta functions
(9.124)
ga
ga 
1 ga

s 


2
1 g2a  s
2
1 s ga

   1 s ga 
Heating Rates
Net flux divergence equation
(9.125)
1  dQsolar dQir 
1
Fn
 T 
  


 
 t  r cp,m  dt
dt  cp,ma z
Net downward minus upward radiative flux
Fn 

0 F  F  d
Partial derivative term
Fn,k

z
(9.126)
 F ,k1 2 F  ,k1 2  F ,k1 2 F  ,k1 2 
zk1 2  zk1 2
Temperature change
(9.127)
Fn
Tk 
h
c p,ma z
1
Photolysis Coefficients
Photolysis rate (s-1) at bottom of layer k
J q, p,k 1 2 
(9.128)

0 4I p,,k 1 2ba,g,q,,T Yq, p,,T d
Radiance at bottom of layer k (photons cm-2 m sr-1 s-1) (9.129)
 10 m3  
I p,,k 1 2  I  ,k 1 2  I ,k 1 2 10

2
cm m  hc



Example 9.14:
--->
--->
I = 12 W m-2 in band 0.495 m <  < 0.505 m
mean = 0.5 m
Ip, = 3.02 x 1015 photons cm-2 s-1