Post on 19-Dec-2015
Atmospheric scatterers
Air molecules~0.0004 µm
Most aerosol(>0.01 µm)
Cloud drops(typically 5-10 µm)
Rain drops
Ice crystals(hail, etc. greater)
Wavelength Frequency
Coarse aerosol(sand, dust
sea salt)
size
<>≈ wavelength
How we can describe radiation Direction, wiggliness, polarization, radiative quantities (e.g., flux, radiance, albedo)
surface reflection, concept of extinction, radiative transfer equation
Direction = zenith angle= azimuth (from North to East)u = cos()µ = |u|Subscript 0: radiation coming from Sun
If interested in not a single specific direction: solid angle ()
surface
radius2
For entire sphere:
4r2
r24
(steradian, unitless)
Wiggliness
Wavelength (): µm (10-6 m), nm (10-9 m), A (10-10 m)
Wavenumber = 1/mof waves in unit dist.
Frequency () = c/sHz (Hertz)of waves passing a point in 1 s
c = 3.108 m/s (speed of light)
Amplitude(A) (not used very often)
Energy (E): W (E ~ A· )
˜
Wavelength Frequency
Radiative quantities
Wm-2µm-1
F F
Flux or irradiance (F): total energy of radiation crossing a surface
Broadband flux: Wm-2
Spectral flux:
F F
Wm-2Hz-1
Radiance or intensity (I): energy of radiation crossing a surface in a particular direction
Broadband radiance: I Wm-2sr-1
Spectral radiance: Wm-2sr-1 µm-1
Spectral radiance: Wm-2sr-1Hz-1
I I
I I
Radiation at surface
Surface of the Earth:
AE 4RE2
Global average irradiance = S0/4 (or F0/4)
F E intercepted
A
F0 Acos0
AF0 cos0
Consequences in weather and climate?• D• S• L
Radiation at surface (continued)
Since flux is integral of intensity:
Downward flux:
F I d I , 0
/ 2
0
2
2 sin cos d d I ,
0
1
0
2
udud
Upward flux:
F I d I , / 2
0
2
2 sin cos d d I ,
1
0
0
2
udud
Albedo ():
0 Freflected
FincomingAlbedo values for natural surfaces (%)
Fresh, dry snow: 70-90Old, melting snow: 35-65Sand, desert: 25-40Dry vegetation: 20-30Deciduous forest: 15-25Grass: 15-25Ocean (low sun): 10-70Bare soil: 10-25Coniferous forest: 10-15Ocean (high sun): < 10
For isotropic radiation (intensity same in all directions):(real-life experience)
F F I
We used above that
du d cos sin dand that
d sin d d
The extinction law
Extinction Law
• The extinction law can be written as
dsIkdI )(
• The constant of proportionality is defined as the extinction coefficient. k can be defined by the length of the absorbing path with the gas at one atmosphere pressure
)()( 1 mdsI
dIk
Optical depth
• Normally we are interested in the total extinction over a finite distance (path length)
s s s
nms nkdskdskds0 0 0
)(')(')(')(
Where S() is the extinction optical depth
• The integrated form of the extinction equation becomes
)(exp),0(),( sIsI
Extinction = scattering + absorption
• Extinction really consists of two distinct processes, scattering and absorption, hence
)()()( ascs
)',()(
)',()(
0
0
sds
sds
i
sii
a
i
sii
sc
where
Differential equation of radiative transfer
• We must now add the process called emission.
• We introduce an emission coefficient, jν• Combining the extinction law with the definition of the
emission coefficient
dsjdsIkdI )(
noting that:
)(
)(
k
jI
d
dI
ddsk
s
s
Differential equation of radiative transfer
• The ratio j/k() is known as the source function,
)(
k
jS
SI
d
dI
s
This is the differential equation of radiative transfer
Scattering
• Two types of scattering are considered – molecular scattering (Rayleigh) and scattering from aerosols (Mie)
• The equation for Rayleigh scattering can be written as
nRAY ()
83
2
4
p2
• Where α is the polarizability
Differential Equation of Radiative Transfer
• Introduce two additional parameters. B, the Planck function, and a , the single scattering albedo (the ratio of the scattering cross section to the extinction coefficient).
• The complete time-independent radiative transfer equation which includes both scattering and absorption is
4
)ˆ,'ˆ('4
)()()(1 Ipd
aTBaI
d
dI
s
Solution for Zero Scattering
• If there is no scattering, e.g. in the thermal infrared, then the equation becomes
dI
d S
I B (T)
Transmittance
• For monochromatic radiation the transmittance, T, is given simply by
/);( eT
• But now we must consider how to deal with radiation that is not monochromatic. In this case the integration must be made over all frequencies.
• Absorption cross section at high spectral resolution are available in tabular form – HITRAN.
• But usually an average value over a frequency interval is used.