Radiation in the atmosphere

Transcription

1 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 λ 1 and λ 2 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. 1

2 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!! 2

3 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 B λ 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 3

4 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 4

5 The Earth as a black body Total solar irradiance Data from PMOD WRC 5

6 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 Q 0 thus is given by: where R m 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 W/m 2 Maximum values of approx. 550 W/m 2 are reached at the poles 6

7 Daily average solar flux at the top of the atmosphere Insolation over the year 2006 as measured at Bern 7

8 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 k a is called the absorption coefficient and has dimensions of m -1 k a is proportional to the number density n where σ a is the absorption cross section k a can also be expressed as mass absorption coefficient k abs : Where the density is known from the gas law 8

9 Absorption coefficient, opacity, transmittance and absorptivity The absorption coefficient is the key parameter: Every constituent contributes in its own way Note the relation of k a with the imaginary part of the index of refraction n i and the dielectric constant ε Integration of the Beer-Lambert law leads to: Opacity τ --> Penetration depth of radiation is 1/ k a Transmittance T value between 0 and 1 Absorptivity A Penetration depth of UV radiation in the atmosphere 9

10 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 τ 0 --> Maximum of energy deposition A maximum of the absorption rate r is reached at the altitude z max If the sun is in zenith this reduces to z max can thus be expressed by z 0 : For the absorption rates r max and r 0 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: 10

11 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: 11

12 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 N 0 = i : 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. 12

13 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 13

14 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 (NH 4 ) 2 SO 4 aerosol at 80% relative humidity for several particle sizes at a wavelength of 550nm 14

15 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 H 2 O is mainly located in the troposphere O 3 primarily in the stratosphere CO 2 has constant VMR 15

16 Radiative Transfer Equation Consider a plane parallel atmosphere. A beam with intensity (radiance) I ν 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 di ν along the path through the layer will be equal to the emission E ν of the gas minus the absorption A ν Absorption is described by the Beer-Lambert law: Where k ν is the mass absorption coefficient: Emission is given by the Planck function and the corresponding emissivity ε ν : 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 τ ν (z): Written in more compact form (leaving away dependencies): 16

17 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 17

18 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 18

19 Longwave heating rates where Primitive model of a single layer atmosphere Consider an atmosphere consisting just of one layer at a temperature T a. The absorbtivity is different for solar radiation and long wave radiation, i.e. a sw and a lw Atmospheric layer situated above surface with temperature T s 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 19

20 Primitive model of a single layer atmosphere Consider an atmosphere consisting just of one layer at a temperature T a. The absorbtivity is different for solar radiation and long wave radiation, i.e. a sw and a lw Atmospheric layer situated above surface with temperature T s 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 T a. The absorbtivity is different for solar radiation and long wave radiation, i.e. a sw and a lw Atmospheric layer situated above surface with temperature T s 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 a sw and a lw both equal 0 : Assume a sw and a lw both not 0 and A=0: 20