Download Radiation in the atmosphere

Radiation in the atmosphere
• Flux and intensity
• Blackbody radiation in a nutshell
• Solar constant
• Interaction of radiation with matter
• Absorption of solar radiation
• Scattering
• Radiative transfer
Irradiance and radiance
Flux and intensity
A radiation field is characterized by its flux, F, and intensity, I.
Flux density or flux (for short) is a measure of the total energy per unit time (power) per
unit area transported by the radiation field through a plane or deposited on a surface.
Flux is expressed in units of Watts per square meter: Wm-2
Flux makes no distinction concerning where the radiation is coming from
Flux is a broadband quantity including radiation between some limits λ1 and λ2
Another expression for flux is irradiance
A radiation field at a given location is completely characterized when we know the flux
and also the direction from where radiation is coming or going. The radiant intensity or
radiance tells in detail the strength and direction of the radiation field.
There is a link between irradiance and radiance or flux and intensity:
The flux on a surface is obtained by integrating the contributions of intensity from all
possible directions visible from that surface.
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Relationship between flux and intensity
Flux density of radiation in direction of Ω through a
surface element dA is proportional to cosθ
For the upward directed flux we get:
For the downward directed flux we get:
If intensity is isotropic, i.e. I is constant over all directions, then:
The net flux is defined as the difference
between upward and downward directed fluxes:
Blackbody radiation in a nutshell
Every object with a temperature T will generally emit radiation at all possible wavelength.
For any particular wavelength λ there is an upper bound on the amount of that radiation.
This upper bound of emitted radiance, i.e. power per square meter per solid angle for T
and λ is given by the so called Planck function:
Total intensity of emitted radiation contributed by the
wavelength interval from λ to λ + d λ
Units usually are W m-2 µm sr-1
The Planck function can also be expressed for a frequency interval from ν to ν + d ν
CAVE!!
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Blackbody radiation in a nutshell cont.
The Planck law can also be expressed as the amount of photons per square meter
and wavelength interval and solid angle. This is useful for photochemical
applications:
By taking the derivative of Bλ and setting to zero the maximum of the radiation can be
determined. This leads to the Wien law:
CAVE!!
When expressed in frequency, the maximum is at a different location.
Remember that the Planck function is a density function
The total flux emitted by a black body over all wavelengths and in the half space is:
This is the Stefan-Boltzmann law, with
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Kirchhoff’s law
Planck’s function describes emission by a black body. This corresponds to the maximum
possible emission from the object.
Real surfaces deviate from the ideal of a blackbody.
The ratio of what is emitted to what actually would be emitted is called emissivity.
There are two cases of interest: emissivity at a single wavelength and over a broad range
of wavelengths
Monochromatic emissivity:
Broadband emissivity:
Emissivity related to Stefan Boltzmann or graybody emissivity:
Kirchhoff’s law relates absorptivity and emissivity:
The sun as a black body
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The Earth as a black body
Total solar irradiance
Data from PMOD WRC
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Daily average solar flux at the top of the atmosphere
What is the total insolation (energy per unit area) at the top of the atmosphere at a given
location over a day, i.e. from sunrise to sunset?
The solar flux on a unit area under a zenith angle θ is
The zenith angle is a function of the geographical location, the date and time.
From astronomy:
Where δ is the declination, ϕ the latitude and h the hour angle
For sunrise and sunset h = H and θ = π / 2, at noon time θ = ϕ - δ.
The length of the day in radian is then
Insolation on top of atmosphere
The insolation Q0 thus is given by:
where Rm is the mean distance of the Earth from the sun and R is the actual distance.
This leads to:
The combined effects of the length of the day, of the variation in the zenith angle,
and the slight change in the distance of the Earth from the sun give characteristic
values of the daily average solar flux on top of the atmosphere.
The annual average is roughly 200 - 400 W/m2
Maximum values of approx. 550 W/m2 are reached at the poles
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Daily average solar flux at the top of the atmosphere
Insolation over the year 2006 as measured at Bern
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Interaction mechanisms of radiation with matter
Absorption of solar radiation on its path through the
atmosphere
Radiation is passing through a layer of thickness
dz under an angle χ
χ is identical to the zenith angle θ
Going through the layer the flux is attenuated by
the amount dI along the path ds
Law of Beer-Lambert
ka is called the absorption coefficient and has dimensions of m-1
ka is proportional to the number density n
where σa is the absorption cross section
ka can also be expressed as mass absorption coefficient kabs:
Where the density is known from the gas law
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Absorption coefficient, opacity, transmittance and
absorptivity
The absorption coefficient is the key parameter:
Every constituent contributes in its own way
Note the relation of ka with the imaginary part of the index of refraction ni and the dielectric constant ε
Integration of the Beer-Lambert law leads to:
Opacity τ
--> Penetration depth of radiation is 1/ ka
Transmittance T
value between 0 and 1
Absorptivity A
Penetration depth of UV radiation in the atmosphere
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Flux in a specific altitude z and energy deposition
Fill in what we know:
• ds expressed by zenith angle θ
• opacity τ by number density n and absorption cross section σ
• n from hydrostatic equilibrium
I∞ = intensity outside atmosphere
The rate of energy deposition is given by
... what can be expressed by filling in, and by the use of the total opacity of the atmosphere τ0
-->
Maximum of energy deposition
A maximum of the absorption rate r is reached at the altitude zmax
If the sun is in zenith this reduces to
zmax can thus be expressed by z0:
For the absorption rates rmax and r0 for zenith case, we finally obtain:
We introduce the dimension less variable Z:
And finally arrive at the energy deposition rate relative to the maximum value
for the sun in zenith direction:
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Chapman layer
Radiation that is absorbed when penetrating the atmosphere leads to layers of energy
deposition. These layers are called Chapman layers and the function describing this
situation is the Chapman function.
There exist different layers in the atmosphere: ozone layer, sodium layer, ionospheric
layers, temperature in the stratosphere etc.
The peak of the absorption occurs at the altitude z for which the optical path, measured in
the direction of incidence from the top of the atmosphere, equals one, i.e.
Scattering of radiation
In addition to absorption light may also be scattered by air molecules, cloud dropletsand aerosols.
Scattering is a redistribution of radiation in different directions. Radiation in the original direction is
diminished and shows up in other directions. This redistribution is characterized by the so called phase
function.
In analogy to absorption a scattering cross section is introduced.
The combined effect of absorption plus scattering is called extinction.
In analogy to geometrical optics one would call the scattering cross section a kind of shadow.
However this “shadow” can be much bigger than the actual geometrical cross section.
The ratio of the scattering cross section to the geometrical area A is called scattering efficiency:
In analogy an extinction efficiency is defined:
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Why scattering is important
Particles of diameters less than approx. 1µm are highly effective at scattering incoming solar radiation.
These particles reduce the amount of incoming solar energy as compared with that in their absence and
consequently cool the Earth.
Mineral dust particles can scatter and absorb both incoming and outgoing radiation. In the visible part,
light scattering dominates and they mainly cool. In the infrared region, mineral dust acts like an absorber
and acts like a greenhouse gas, thus warms.
Sulfate aerosols and smoke of biomass burning are currently estimated to exert a global average
cooling effect.
Aerosol concentrations are highly variable in space and time.
Greenhouse gas forcing operates day and night. Whereas aerosol forcing due to scattering operates
only during daytime.
Aerosol radiative effects depend in a complicated way on the solar angle, relative humidity, particle size
and composition and the albedo of the underlying surface.
For the interaction of solar radiation with atmospheric aerosols, elastic light scattering is the process of
interest.
The absorption and elastic scattering of light by a spherical particle is a classical problem in physics, the
mathematical formalism of which is called Mie theory.
Aerosols influence climate directly by scattering and absorption of solar radiation and indirectly
through their role as cloud condensation nuclei.
Key parameters used in describing scattering
Key parameters are:
• the wavelength
• the particle size in relation to the wavelength
• the complex index of refraction
The refractive index is normalized to the one of air N0=1.00029+0i :
The distribution of the scattered radiation as a function of the scattering angle is give by the
phase function
The determination of the phase function and the scattering efficiency is mathematically
difficult. Closed theories are only available for the most simple cases.
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Scattering regimes depending on particle
size and wavelength
Rayleigh scattering
Scattering of solar radiation on air molecules belongs to the Rayleigh scattering regime.
The phase function is given by
Rayleigh scattering is symmetrical in forward and backward directions
The phase function depends on polarization of incoming light
Electrical field normal to scattering plane
Electrical field parallel to scattering plane
Unpolarized case
Effect visible with polarizing sunglasses
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Scattering efficiencies
In the Rayleigh regime, i.e. for particles with a diameter of less than approx. 0.1µm and
visible radiation, it can be shown:
Rayleigh scattering is proportional to
--> Blue sky, red sunsets
Absorption is proportional to
For particles with α >> 1 (geometric scattering regime), e.g. water droplets, it can be shown
Scattering function of light for aerosol particles of
different sizes
(NH4)2SO4 aerosol at 80% relative humidity for several particle sizes at a wavelength of 550nm
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Schematic relationship between backscatter and upscatter
fraction of solar radiation
Note: It is the upward hemisphere that is important in the effect of aerosols on Earth’s
radiative balance, not the back hemisphere relative to the direction of incident radiation
Shortwave radiative heating rate in K/day
The heating experienced by a layer of air due to radiation transfer can be expressed in terms of the
rate of temperature change
H2O is mainly located in the
troposphere
O3 primarily in the stratosphere
CO2 has constant VMR
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Radiative Transfer Equation
Consider a plane parallel atmosphere. A beam with intensity (radiance) Iν is propagating upwards
through a layer with thickness dz making an angle θ with the vertical direction.
Layer will absorb but due to the Kirchhoff law will also emit.
The change of intensity dIν along the path through the layer will be equal to the emission Eν of the gas
minus the absorption Aν
Absorption is described by the Beer-Lambert law:
Where kν is the mass absorption coefficient:
Emission is given by the Planck function and the corresponding emissivity εν:
The emissivity is given by Kirchhoff:
Schwarzschild equation:
Radiative Transfer Equation cont.
With the opacity
The Schwarzschild-equation can be expressed
( ) show the dependence of the radiance on zenith angle and altitude.
Altitude is represented parametrically by vertical optical depth.
Multiplication on both sides with e-τ leads to a form that can be integrated:
Integration from the surface, where the optical depth is zero, to some altitude where we
wish to calculate the upward intensity and where the opacity is τν(z):
Written in more compact form (leaving away dependencies):
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Radiative Transfer Equation cont.
Interpretation:
The first term represents the emission of the surface (where the opacity is τ when looking
down from the point of observation), reduced by the absorption along the path from the
surface to the “sensor”.
The second term represents the summation of the emission from all of the atmospheric layers
below the observation point, taking into account subsequent attenuation on the way to the
“sensor”
Almost all radiative transfer problems involving emission and absorption, without scattering,
can be solved with this equation.
... But sometimes that gets very complex !!
Radiative Transfer Equation cont.
The total flux of upward radiation could in principle be obtained by integrating over all
frequencies, i.e. over thousands and thousands of transitions, and over all directions.
This is done by so called line by line calculations based on spectral data bases, such as
• HITRAN
• MODTRAN
• LOWTRAN
Line by line calculations are not easily available and approximations are thought for, e.g. with
so called “band models” and where average flux transmissions are defined
Leading to a net flux
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Fluxes on top of the atmosphere and on the surface
Discussion:
• Main contribution to the integrals from levels where transmittance is changing most rapidly
• In case of outgoing radiation, only a small amount of the emission of the surface is able to escape
• Under typical conditions most of the outgoing radiation originates in the troposphere where
temperatures are lower than on the surface
• When an absorbing atmosphere is present, the average emission temperature is less than the
surface value, and the loss of energy by emission to space is less than the infrared emission from the
surface --> greenhouse effect
• The downward flux on the surface originates in the lower troposphere where most of the water
vapor is. On global average this is almost double the amount than coming from the sun
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Longwave heating rates
where
Primitive model of a single layer atmosphere
Consider an atmosphere consisting just of one layer at a temperature Ta. The absorbtivity is different for
solar radiation and long wave radiation, i.e. asw and alw
Atmospheric layer situated above surface with temperature Ts and albedo A for solar radiation and
emissivity of ε=1 for long wave emission
Consider fluxes at different positions to find temperature in thermal equilibrium
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Primitive model of a single layer atmosphere
Consider an atmosphere consisting just of one layer at a temperature Ta. The absorbtivity is different for
solar radiation and long wave radiation, i.e. asw and alw
Atmospheric layer situated above surface with temperature Ts and albedo A for solar radiation and
emissivity of ε=1 for long wave emission
Consider fluxes at different positions to find temperature in thermal equilibrium
Equilibrium case:
Primitive model of a single layer atmosphere
Consider an atmosphere consisting just of one layer at a temperature Ta. The absorbtivity is different for
solar radiation and long wave radiation, i.e. asw and alw
Atmospheric layer situated above surface with temperature Ts and albedo A for solar radiation and
emissivity of ε=1 for long wave emission
Consider fluxes at different positions to find temperature in thermal equilibrium
Equilibrium case:
Assume asw and alw both equal 0 :
Assume asw and alw both not 0 and A=0:
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