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  • 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
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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

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Page 1: 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

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

Page 2: 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

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)

Page 3: 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

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

Page 4: 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

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

Page 5: 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

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

Page 6: 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

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

Page 7: 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

The extinction law

Page 8: 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

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

Page 9: 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

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

Page 10: 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

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

Page 11: 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

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

Page 12: 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

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

Page 13: 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

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

Page 14: 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

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

Page 15: 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

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)

Page 16: 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

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.


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