Module 9 Radiative Transfer in the Atmosphere 9.1 Introduction It has been emphasized in earlier lectures that energy from Sun drives the circulation of the atmosphere and ocean. Earth receives energy from Sun as ultraviolet, visible and near-infrared radiation. This radiation band is called the short wave radiation or radiation with wavelengths λ < 4 µm . An equal amount of energy is re-emitted by earth to maintain an overall energy balance. Earth emits energy in the form of infrared thermal radiation or longwave radiation with wavelengths λ > 4 µm . These two bands shown in Fig. 9.1 pertain to the wavelength intervals given as, Band 1: Solar spectrum 0.1− 4 µm (1 µm = 10−6 m) Band 2: Infrared region 4 −100 µm Sun Earth Fig. 9.1 Blackbody emission at the temperatures of Sun and Earth. 6000 K 288 K Intensity Overlap region Overlap 0.1 0.5 1 5 10 50 100 Wave length ( µ m ) Microwave radiation is not important for energy balance of the earth, but it finds wide use in remote sensing of the earth system because it is capable of penetrating through clouds. The study of radiative transfer in atmosphere and ocean is important because it leads to (i) a better understanding of energy transfer in these two components of the climate system; and (ii) in interpreting remote sensing measurements from satellites. Radiation is described by flux, intensity and radiance; and there are certain terms also that are frequently used while discussing the mathematics of the radiative transfer. First of all, it is required to calculate a differential amount of radiant energy dEλ ( J ) in a wavelength ( µm ) interval λ and λ + dλ that crosses an area ( m2 ) element dA in time (seconds) interval dt in a direction confined by an arc dω of the solid angle (steradian abbreviated as sr ). Some frequently used terms in radiative transfer are defined below and their dimensions (given in brackets) indicate how they are interrelated. Radiant Energy: The energy of radiation beam at a given wavelength (J) . Flux: Radiant energy crossing unit area per unit time ( Js−1m−2 or Wm−2 ). Intensity: Flux in unit wavelength and unit solid angle (W m−2µm−1 sr−1 ) Flux density or irradiance: Normal component of intensity integrated over the hemispheric solid angle (W m−2µm−1 ). It is the emittance of an emitting surface. Extinction: Loss of energy from a photon due to absorption and scattering. 1 Radiative decay: Electron falls back to the original state by re-emitting photon of same energy and same frequency. Photodissociation: Absorption of solar photons leading to breakdown of molecules initiating photochemical reactions and photo-ionization (i.e. outer electrons are stripped from atoms). Absorption: At sufficiently high pressures, molecular collisions are likely to occur and as a result of molecular collisions energy of excitation will be transferred to other forms of energy before re-emission could take place. In such situations, the photon is said to have been absorbed. Due to absorption of energy of excitation, the kinetic energy is produced, which is shared between molecules by collisional interaction. Since thermal energy is the macroscopic expression of molecular kinetic energy, the increased kinetic energy of molecules leads to local heating. In this process the photon energy has been transferred to heat. This process is also called “thermalization” or “quenching”. Scattering: Scattering of a photon of given energy and frequency by an atmospheric molecule implies that electron falls back to ground state re-emitting photon of same energy and frequency as the original one but in random direction. There are two kinds of scattering that occur in the atmosphere. Scattering by air molecules (Rayleigh) gives the blue colour to the sky; and scattering by aerosol particles (such as dust), called the Mie scattering, produces most scenic sunsets or sunrises. Thermal (infra-red) cooling: Thermal photons are also absorbed or scattered much like solar photons. When emitted, the energy is drawn from molecular kinetic energy leading to cooling of the atmosphere. Thus, radiant energy dEλ and the intensity Iλ are related as follows: dEλ = Iλ dAdλ dω dt (9.1) 9.2 Absorbing constituents in the atmosphere The main constituents that absorb radiation in different wavelength regions (or bands) are shown in Fig. 9.2. One may conspicuously note in this figure two atmospheric windows where the absorption is weak: one in the solar spectrum of the wavelengths 0.3 < λ < 1 µm (visible) and the other in the infrared region of wavelengths 8 < λ < 12 µm . Ozone (O3), water vapour (H2O) and carbon dioxide (CO2) are the dominant absorbers that primarily heat the atmosphere though oxygen (O2) also absorbs radiation in the ultraviolet and the visible region of the electromagnetic spectrum. In the solar spectrum ( λ < 4 µm ), O3 absorbs radiation in the ultraviolet and visible region; both CO2 and water vapour are greenhouse gases and absorb radiation in the near infrared region 1 < λ < 3.5 µm . In the infrared region, radiation is weakly absorbed by CO, CH4 and N2O in the troposphere, but CO2 and water vapour are the strong absorbers. In the atmospheric window region, 8 < λ < 12 µm , absorption is weak by these gases except for a band near λ = 9.6 µm which is associated with ozone (O3). The atmospheric window regions in the visible and infrared wavelength bands are important for remote sensing of atmosphere; that is why satellites are regularly launched by different countries in the space and their constellation forms an important part of the earth observing system. Electromagnetic radiation, a packet of different waves, is characterized by wavelength, frequency and wave number. Wavelength is denoted by λ; wave number is the reciprocal of wavelength; frequency is the product of wave number and the speed of 2 light with which all electromagnetic waves propagate. Wavelength (λ), wave number (ν~) and frequency (ν) are thus related as: 1 ν c ν = = and ν = cν~ = , c is the speed of light. λ c λ Fig. 9.2: The electromagnetic spectrum and the main atmospheric absorbents Small variations in the speed of light in air produce mirage and distortions, which indeed limit the resolution of ground based telescopes. Moreover, differences in the speed of light in air and water produce beautiful rainbows. Wavelength is expressed in the units of micrometre ( µm ). Based on wavelength (or frequency or wave number), energy can be partitioned according to wavelength bands. However, for radiative transfer in the atmosphere, radiation is categorized into two wave bands: Shortwave radiation: λ < 4 µm energy associated with solar (downwelling) radiation Longwave radiation: λ > 4 µm energy associated with terrestrial (upwelling) radiation Monochromatic radiation: refers to radiation of one single frequency or wavelength (i.e. single colour radiation). Similar meaning is also ascribed to monochromatic radiance. Visible Region: 0.39 ≤ λ ≤ 0.76 µm ; that is, the wavelengths sensed by the eye. Photon: Discrete packets of radiation; each photon contains energy E = hν . 9.3 Black-body radiation It is the radiation emitted from a small hole that has been cut in an isothermal cavity whose walls are maintained at a uniform temperature. The radiation is isotropic and the spectral energy density (the energy per unit volume per unit frequency interval) depends only on one frequency and cavity wall temperature. Thus, a blackbody surface completely absorbs all incident radiation to it. Most caves appear nearly black as all the sunlight rays that enter the cave are absorbed in multiple reflections and only a small fraction of sunlight (incident) emerges out from the entrance. The Planck function: According to the Planck’s Law (the first fundamental law of radiation), spectral energy density of the blackbody radiation at temperature T is given by 8π hν 3 u (T ) = (9.2) ν c3 {exp[hν / (kT )]−1} 3 −34 Here uν (T ) is the spectral energy density of blackbody radiation; h = 6.6262 ×10 J s 23 1 (Planck constant), k = 1.38062 ×10− J K − (Boltzmann constant) and c is the speed of 8 −1 light (c = 2.99792458 ×10 ms ). The energy density of a group of photons moving within a small arc of the solid angle Δω steradians is given by uν Δω / (4π ) ; 4π is the solid angle of the sphere. Then the energy flow at speed c by this group of photons is given as ⎛ uν Δω ⎞ c uν c ⎜ ⎟ = = Bν (T ) (9.3) ⎝ 4π ⎠ Δω 4π Using the expression of uν (T ) , we obtain an expression for the Planck function Bν (T ) as 2hν 3 B (T ) = (Planck Function) (9.4) ν c2 {exp[hν / (kT )]−1} Bν (T ) is the power per unit area, per unit solid angle, per unit frequency interval (spectral radiance) of blackbody radiation. The Planck function may also be expressed as a function of wavelength λ and temperature T, and it reads as: 2hc2 B (T ) = Wm−3 sr−1 or Wm−2sr−1µm−1 . (9.5) λ λ 5 exp[hc / (λkT )]−1 ( ) { } Since h, c and k are constant, the intensity of radiation Bλ(T) may be written as: −5 c1λ 2 −16 2 Bλ (T ) = ; c1 = 2πhc = 3.74 ×10 W m ; c2 π exp −1 c = hc / k= 1.45 ×10−2 mK. (9.6) { λT } 2 Thus Bλ (T ) is the blackbody spectral radiance written in terms of power (W ) per unit area (m−2 ) , per unit solid angle (sr−1) , per unit wavelength interval (m−1 or µm−1) . Whenever radiation and matter interact; photons are absorbed, scattered or emitted by molecules of the optically active gas If Bλ (T ) is integrated over all wavelengths, then one obtains the following expression for the blackbody radiance, ∞ σ 2k 4π 5 B (T )dλ = T 4 , σ = = 5.67 ×10−8 Wm−2K −4 (9.7) ∫ λ 3 2 0 π 15h c ∞ x3 π 4 In evaluating the above integral we use, dx = .