Atmospheric scatterers Wavelength Frequency Rain drops Ice crystals (hail, etc. greater) Cloud drops (typically 5-10 µm) Coarse aerosol (sand, dust sea salt) < size ≈ wavelength Most aerosol (>0.01 µm) > Air molecules ~0.0004 µm

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 () For entire sphere: (steradian, unitless)

Wiggliness Wavelength Frequency Wavelength (): µm (10-6 m), nm (10-9 m), A (10-10 m) Wavenumber = 1/mof waves in unit dist. Frequency () = c/sHz (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· )

Radiative quantities Flux or irradiance (F): total energy of radiation crossing a surface Broadband flux: Wm-2 Spectral flux: Wm-2Hz-1 Wm-2µm-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

Radiation at surface Consequences in weather and climate? D S L Surface of the Earth: Global average irradiance = S0/4 (or F0/4)

Radiation at surface (continued) Since flux is integral of intensity: Downward flux: Upward flux: We used above that and that For isotropic radiation (intensity same in all directions): (real-life experience) Albedo (): Albedo values for natural surfaces (%) Fresh, dry snow: 70-90 Old, melting snow: 35-65 Sand, desert: 25-40 Dry vegetation: 20-30 Deciduous forest: 15-25 Grass: 15-25 Ocean (low sun): 10-70 Bare soil: 10-25 Coniferous forest: 10-15 Ocean (high sun): < 10

The extinction law

Extinction Law The extinction law can be written as 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

Optical depth Normally we are interested in the total extinction over a finite distance (path length) Where tS(n) is the extinction optical depth The integrated form of the extinction equation becomes

Extinction = scattering + absorption Extinction really consists of two distinct processes, scattering and absorption, hence 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 noting that:

Differential equation of radiative transfer The ratio jn/k(n) is known as the source function, 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 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

Solution for Zero Scattering If there is no scattering, e.g. in the thermal infrared, then the equation becomes

Transmittance For monochromatic radiation the transmittance, T, is given simply by 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.