Scattering and Absorption (PDF)

Scattering is a fundamental physical process that takes place within the atmosphere. It is associated with light and its interaction with matter. It removes energy from a beam of light traversing the medium, i.e. the beam of light is attenuated. Attenuation is also called extinction. Atmosphe...

Scattering is a fundamental physical process that takes place within the atmosphere. It is associated with light and its interaction with matter. It removes energy from a beam of light traversing the medium, i.e. the beam of light is attenuated. Attenuation is also called extinction. Atmospheric scattering occurs at all wavelengths throughout the entire electromagnetic spectrum. It is a process, by which a particle in the path of an electromagnetic wave continuously detracts energy from the incident wave and redirects that energy in all directions. When a particle at a given position removes the incident light by scattering just once in all directions, we call it single scattering. A portion of this scattered light, of course, may reach another particle, where it is scattered again in all directions (secondary scattering). Scattering more than once is called multiple scattering. Scattering of light by particles depends on particle shape, particle size, particle index of refraction, viewing geometry, and the wavelength of radiation Light is said to be reflected when the angle, at which light initially strikes a surface, equals to the angle at which light bounces off the same surface. While in the terrestrial spectral range, absorption and emission play the dominant role, scattering prevails in the solar spectrum at cloud-free conditions.  Scattering is also a selective process by which light is scattered by Air molecules (10−4 𝜇𝑚), Aerosol particles (1 𝜇𝑚), Hydrometeors – Water droplets (10 𝜇𝑚), – Ice crystals (100 𝜇𝑚), Schematic view of angular patterns of the radiation at – Large raindrops and hail particles (104 𝜇𝑚) visible (0.5 𝜇𝑚) of scattered intensity from spherical particles of three different sizes (a) small (10−4 𝜇𝑚), (b) large (0.1 𝜇𝑚), and (c) huge (1 𝜇𝑚) particles Scattering size parameter that can be given by  Geometric Optics Geometric optics can explain a wide variety of optical phenomena. For 𝜒 greater or equal to 50, the sphere is large as compared to the wavelength. Under these conditions, the laws of geometrical optics can be applied. The interaction of solar radiation with virtually all types of hydrometeors and of infrared radiation with hydrometeors in general falls into this category. A rainbow results from the reflection of light in raindrops. Sometimes a primary and a secondary rainbow are visible. A secondary rainbow develops when light entering a raindrop undergoes two internal reflections instead of just one. The colors of the primary rainbow go from red on the outside to violet on the inside. The opposite is true for the secondary rainbow. Since the intensity of light is reduced even further by the second reflection, the secondary rainbow appears to be less bright than the primary one. The primary rainbow has always and angle of 40–42 degree from the path of the Sun rays.  Rayleigh Scattering The simplest and in some ways the most important example of a physical law of light scattering with various applications is that discovered by Rayleigh (1871). His findings led to the explanation of the blue color of the sky. Consider a small homogeneous, isotropic, spherical particle whose radius is much smaller than the wavelength of the incident radiation. The incident radiation produces a homogeneous electric field E0, called the applied field. Since the particle is very small, the applied field generates a dipole configuration on it. The electric field of the particle, caused by the electric dipole, modifies the applied field inside and near the particle. Let E be the combined field, and p0 be the induced dipole moment then p0 = αE0. Where α is the polarizability of the particle. The applied field E0 generates oscillation of an electric dipole in a fixed direction. The oscillating dipole, in turn, produces a plane-polarized electromagnetic wave, the scattered wave. Let r is the distance between the dipole and the observation point 𝛾 is the angle between the scattered dipole moment p and the direction of observation, c is the velocity of light 1 1 𝜕2 𝐏 According to classical theory by Hertz 𝐄 = sin 𝛾 c2 r 𝜕t2 𝜕2 𝐏 Where is the acceleration of the scattered dipole moment. 𝜕t2  In an oscillating periodic field, the scattered dipole moment 𝐏 may be written in terms of the induced dipole moment 𝐏0 as 𝐏 = 𝐏0 𝑒 −𝑖𝑘(𝑟−𝑐𝑡) 2𝜋 Where k is the wavenumber 𝑘 = , and kc = ω is the angular frequency. 𝜆 𝝏𝑷 = 𝐏0 𝑒 −𝑖𝑘 𝑟−𝑐𝑡 (𝑖𝑘𝑐) 𝝏𝒕 𝜕2𝐏 2 = 𝐏0 𝑒 −𝑖𝑘 𝑟−𝑐𝑡 (𝑖𝑘𝑐)2 𝜕t 1 1 𝜕2 𝐏 So, 𝐄 = sin 𝛾 give, c2 r 𝜕t2 𝑬0 𝑒 −𝑖𝑘 𝑟−𝑐𝑡 𝛼𝑘 2 sin 𝛾 𝐄=− 𝑟 Let the scattered electric field of sunlight be decomposed into 𝐸𝑟 and 𝐸𝑙 , perpendicular and parallel to the plane of scattering respectively. In this case, we may consider separately the scattering of the two electric field components E0r and E0l by molecules assumed to be homogeneous, isotropic, spherical particles. 𝑬0𝑟 𝑒 −𝑖𝑘 𝑟−𝑐𝑡 𝛼𝑘 2 sin 𝛾1 𝑬𝒓 = − 𝑟  𝑬0𝑙 𝑒 −𝑖𝑘 𝑟−𝑐𝑡 𝛼𝑘 2 sin 𝛾2 𝑬𝒍 = − 𝑟 In matrix form, we may write 𝜋 𝜋 Where 𝛾1 = , 𝛾2 = − Θ, Θ is the scattering angle. 2 2 Now the intensity components (per solid angle) of the incident and scattered radiation can be define in the forms I0 = C|E0|2 and I = C|E|2, where C is a certain proportionality factor such that C/r2 implies a solid angle. where Ir and Il are polarized intensity components perpendicular and parallel to the plane containing the incident and scattered waves, i.e., the plane of scattering. The total scattered intensity of the unpolarized sunlight incident on a molecule is then 𝐼 For unpolarized sunlight 𝐼0𝑟 = 𝐼0𝑙 = 20, so we obtain This is the original formula derived by Rayleigh, and we call the scattering of sunlight by molecules Rayleigh scattering. By this formula, the intensity of unpolarized sunlight scattered by a molecule is proportional to the incident intensity I0 and is inversely proportional to the square of the distance between the molecule and the point of observation. The scattered intensity also depends on the polarizability, the wavelength of the incident wave, and the scattering angle.  For vertically (r) polarized incident light, the scattered intensity is independent of the direction of the scattering plane. In this case then, the scattering is isotropic. For horizontally (l) polarized incident light, the scattered intensity is a function of 𝑐𝑜𝑠 2 Θ. When the incident light is unpolarized, such as sunlight, the scattered intensity depends on (1 + 𝑐𝑜𝑠 2 Θ) Polar diagram of the scattered intensity for Rayleigh molecules: (1) polarized incident light with the electric vector perpendicular to the scattering plane, (2) polarized incident light with the electric vector on the scattering plane, and (3) unpolarized incident light. Let us define a nondimensional parameter called the phase function 𝑃(𝑐𝑜𝑠Θ) such that 2𝜋 𝜋 𝑃(𝑐𝑜𝑠Θ) න න 𝑠𝑖𝑛Θ𝑑Θ𝑑𝜙 = 1 4𝜋 0 0 3 The phase function for Rayleigh scattering is 𝑃 𝑐𝑜𝑠Θ = (1 + 𝑐𝑜𝑠 2 Θ) 4 𝐼0 2 128𝜋5 𝑃(Θ) So the scattering intensity now become 𝐼 Θ = 𝛼 𝑟2 3𝜆4 4𝜋 the angular distribution of the scattered intensity is directly proportional to the phase function. The scattered flux f (or power, in units of energy per time) can be evaluated by integrating the scattered flux density (I ) over the appropriate area a distance r away from the scatterers. Thus 𝑓 = ‫׬‬Ω (𝐼ΔΩ)𝑟 2 𝑑Ω where 𝑟 2 𝑑Ω represent the area according to the solid angle. 𝑑Ω = 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 𝐼0 2 128𝜋 5 𝑃(Θ) 2 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 = 𝐹 𝛼 2 128𝜋 5 𝑓 = න ( 2𝛼 4 ΔΩ)𝑟 0 Ω 𝑟 3𝜆 4𝜋 3𝜆4 Where 𝐹0 = 𝐼0 ΔΩ is the incident flux. Scattering cross-section per molecule 𝑓𝑠 128𝜋 5 𝜎𝑠 = = 𝛼2 𝐹0 3𝜆4 The scattering cross section (in units of area) represents the amount of incident energy that is removed from the original direction because of a single scattering event such that the energy is redistributed isotropically on the area of a sphere whose center is the scatterer and whose radius is r. The scattered intensity can be written as 𝐼0 𝑃(Θ) 𝐼 Θ = 2 𝜎𝑠 𝑟 4𝜋 This is the general expression for scattered intensity, which is valid not only for molecules but also for particles whose size is larger than the incident wavelength. The polarizability α, which was used in the preceding equations, can be derived from the principle of the dispersion of electromagnetic waves and is given by Let us derive it!!  Complex Index of Refraction, Dispersion of Light, and Lorentz–Lorenz Formula Within a dielectric, positive and negative charges are impelled to move in opposite directions by an applied electric field. As a result, electric dipoles are generated. The product of charges and the separation distance of positive and negative charges is called the dipole moment, which, when divided by the unit volume, is referred to as polarization P. The displacement vector D (charge per area) within a dielectric is defined by  But the polarization vector for N dipoles is where v denotes the velocity of an electron, which is very small compared to the velocity of light. Hence, the force produced by the magnetic field may be neglected. The force in the vibrating system in terms of the displacement r is due to (1) the acceleration of the electron; (2) the damping force, which carries away energy when the vibrating electrons emit electromagnetic waves, and which is proportional to the velocity of the electrons; and (3) the restoring force of the vibration, which is proportional to the distance r. From Newton’s second law, we find Let us consider a dielectric placed between the plates of a parallel-plate condenser without the end effect. Moreover, we consider an individual molecule constituting this dielectric and draw a sphere with radius a about this molecule. The molecule is, therefore, affected by the fields caused by (1) the charges of the surfaces of the condenser plates; (2) the surface charge on the dielectric facing the condenser plates; (3) the surface charge on the spherical boundary of radius a; and (4) the charges of molecules (other than the one under consideration) contained within the sphere. For items (1) and (2), the electric field produced by these charges is  The refractive index is an optical parameter associated with the velocity change of electromagnetic waves in a medium with respect to a vacuum. Normally, the refractive indices of atmospheric particles and molecules are composed of a real part 𝑚𝑟 and an imaginary part 𝑚𝑖 corresponding, respectively, to the scattering and absorption properties of particles and molecules. In the solar visible spectrum, the imaginary parts of the refractive indices of air molecules are so insignificantly small that absorption of solar radiation by air molecules may be neglected in the scattering discussion. The real parts of the refractive indices of air molecules in the solar spectrum are very close to 1, but they depend on the wavelength (or frequency) Because of this dependence, white light may be dispersed into component colors by molecules that function like prisms. The real part of the refractive index may be approximately fitted by Since 𝑚𝑟 is close to 1, for all practical purposes, we can write 1 𝛼= (𝑚𝑟2 − 1) 4𝜋𝑁𝑠 Thus the scattering cross section can be written as 8𝜋 3 𝑚𝑟2 − 1 2 𝜎𝑠 = 𝑓(𝛿) 3𝜆4 𝑁𝑠2 where, 𝑓 𝛿 = (6 + 3𝛿)/(6 − 7𝛿) is the correction factor for the anisotropy of the medium. The optical depth of the entire molecular atmosphere at a given wavelength may be calculated from the scattering cross section in the form 𝑧=∞ 𝜏 𝜆 = 𝜎𝑠 (𝜆) න 𝑁 𝑧 𝑑𝑧 0 Where, 𝑁 𝑧 denotes the number density of molecules as a function of height, and 𝑧 = ∞ is the top of the atmosphere. The optical depth represents the attenuation power of molecules with respect to a specific wavelength of the incident light.  Why does the sky look blue? The scattered intensity depends on the wavelength of incident light and the index of refraction of air molecules contained in the polarizability term. According to the analyses, the index of refraction also depends slightly on the wavelength. Thus, the intensity scattered by air molecules in a specific direction 1 may be symbolically expressed in the form 𝐼𝜆 ~ 4 The inverse dependence of the scattered intensity on the 𝜆 wavelength to the fourth power is a direct consequence of the theory of Rayleigh scattering and is thefoundation for the explanation of blue sky. A large portion of solar energy is contained between the blue and red regions of the visible spectrum. Blue light (λ ≈ 0.425 μm) has a shorter wavelength than red light (λ ≈0.650 μm). Consequently, blue light scatters about 5.5 times more intensity than red light It is apparent 1 that the 𝐼𝜆 ~ 4 law causes more blue light to be scattered than red, green, and yellow, and so the sky, when 𝜆 viewed away from the sun’s disk, appears blue. Moreover, since molecular density decreases drastically with height, it is anticipated that the sky should gradually darken to become completely black in outer space in directions away from the sun. And the sun itself should appear whiter and brighter with increasing height. As the sun approaches the horizon (at sunset or sunrise), sunlight travels through more air molecules, and therefore more and more blue light and light with shorter wavelengths are scattered out of the beam of light, and the luminous sun shows a deeper red color than at its zenith. However, since violet light (∼0.405 μm) has a shorter wavelength than blue, a reasonable question is, why doesn’t the sky appear violet? This is because the energy contained in the violet spectrum is much less than that contained in the blue spectrum, and also because the human eye has a much lower response to the violet color.  Mie Scattering 2𝜋𝑎 The size parameter is defined as 𝑥 = where, a is the radius of the particle, If 𝑥 ≪ 1 then the scattering 𝜆 is Rayleigh scattering and if 𝑥 ≥ 1 the scattering is called Lorenz-Mie scattering. The intensity scattered by a particle as a function of direction, is given by 𝑃(Θ) 𝑃(Θ) 𝐼 Θ = 𝐼0 Ω𝑒𝑓𝑓 = 𝐼0 𝜎𝑠 2, 4𝜋 4𝜋𝑟 where Ω𝑒𝑓𝑓 is the effective solid angle upon which scattering occurs. The Lorenz-Mie scattering cross section by spherical particle can be given by the following expansion: 𝜎𝑠 4 (1 + 𝑐 𝑥 2 + 𝑐 𝑥 4 + ⋯ … … … … …. ) = 𝑄 𝑠 = 𝑐1 𝑥 2 3 𝜋𝑎2 𝑄𝑠 is referred to as the scattering efficiency, and the coefficients in the case of non-absorbing particles are given by The leading term is the dipole mode contribution associated with Rayleigh scattering  GEOMETRIC OPTICS The principles of geometric optics are the asymptotic approximations of the fundamental electromagnetic theory and are valid for light-scattering computations involving a particle whose dimension is much larger than the wavelength, i.e., 𝑥 ≫ 1 A light beam can be thought of as consisting of a bundle of separate parallel rays that hit the particles in this context. Each ray will undergo reflection and refraction and will pursue its own path along a straight line outside and inside the scatterer with propagation directions determined by the Snell law let v1 and v2 be the velocities of propagation of plane waves in the two media such that v1 > v2. Also, let θi and θt be the angles corresponding to the incident and refracted waves. Thus, we have sin θi/ sin θt = v1/v2 = m, where m is the index of refraction for the second medium with respect to the first. In the context of geometric optics, the total electric field is assumed to consist of the diffracted rays, the reflected rays, and the refracted rays, as illustrated in Fig. using a sphere as an example The energy carried by the diffracted and the Fresnelian rays (reflected and refracted rays) is assumed to be the same as the energy intercepted by the particle cross section projected along the incident direction. From the Babinet’s principle (The diffraction pattern in the far field, referred to as Fraunhofer diffraction, from a circular aperture is the same as that from an opaque disk or sphere of the same radius.) Based on this principle and geometric consideration, the scattered intensity can be given by 𝑥 4 2𝐽1 (𝑥 𝑠𝑖𝑛Θ) 2 𝐼𝑃 = 4 𝑥 𝑠𝑖𝑛Θ Where 𝐽1 is the first-order Bessel function and Θ is the scattering angle. If a particle of any shape is much larger than the incident wavelength, the total energy removed is based on geometric reflection and refraction, giving an effective cross-section area equal to the geometric area A. In, addition, according to Babinet’s principle, diffraction takes place through a hole in this area, giving a cross-section area also equal to A. The total removal of incident energy is therefore twice the geometric area. Thus, the extinction cross section is given by 𝜎𝑒 𝜎𝑒 = 2𝐴 or 𝑄𝑒 = =2 𝐴 where 𝑄𝑒 is called the extinction efficiency. If a particle is nonabsorbing, then we have 𝑄𝑒 = 𝑄𝑆 , where the extinction and scattering efficiencies are the same.  ANOMALOUS DIFFRACTION THEORY Consider large optically soft particles such that 𝑥 ≫ 1 and 𝑚 − 1 ≪ 1. In this case, the extinction is largely caused by absorption of the light beam passing through the particle, as well as by the interference of light passing through the particle and passing around the particle, which is he basis of anomalous diffraction. Let the plane wave be incident on a spherical particle with a radius a and a refractive index m → 1. The wave front on the forward side of the particle can be divided into two types: one within the geometric shadow area denoted by A = πa2, and one outside this area. The incident rays can undergo diffraction and pass around the particle. The rays can also hit the particle and undergo reflection and refraction and pass through it. However, these rays will have phase lags due to the presence of the particle. The phase lag for the ray indicated in the figure is 2a sin α(m - 1) · 2π/λ. If we define the phase shift parameter 𝜌 = 2𝑥(𝑚 − 1), the phase lag can be written as 𝜌 sin 𝛼  The resultant wave on the collection screen is the sum of the incident and scattered fields. If the incident field is assumed to be unity, then in the forward direction ( Θ = 0), the change in the electric field is proportional to 4 4 The extinction coefficient 𝑄𝑒 = 4 𝑅𝑒 𝐾 𝑖𝜌 = 2 − 𝜌 sin 𝜌 + 𝜌2 (1 − cos 𝜌)  Absorption Efficiency The ray path as shown in Fig. is l = 2a sin α. The absorption coefficient ki = mi2π/λ, where mi is the imaginary part of the refractive index. --Thus, the absorption path length associated with the electric field is 𝑙𝑘𝑖 --The attenuation of the intensity of the ray is then exp(- 𝑙𝑘𝑖 ) --The absorption cross-section for all possible rays is Absorption efficiency where b = 4xmi and x = 2πa/λ. The approximation based on the anomalous diffraction theory (ADT) is useful for the calculation of the extinction and absorption coefficients when m → 1. It can also be applied to nonspherical particles such as spheroids and hexagons.  Atmospheric Solar Heating Rates The absorption of solar radiation by various gases is important because of its generation of heating in the atmosphere, which is also affected by multiple scattering processes. Consider a plane-parallel absorbing and scattering atmosphere illuminated by the solar spectral irradiance 𝐹⊙ so that the downward flux density normal to the top of the atmosphere is given by 𝜇0 𝐹⊙. Let the differential thickness within the atmosphere be Δ𝑧, and let the spectral downward and upward flux densities centered at wavelength λ be denoted by F↓ and F↑, respectively. The net flux density (downward) at a given height z is then defined by 𝐹 𝑧 = 𝐹 ↓ 𝑧 − 𝐹 ↑ (𝑧) Due to the absorption, the net flux density decreases from the upper levels to the progressively lower levels The net flux density divergence for the differential layer is, ∆𝐹 𝑧 = 𝐹 𝑧 − 𝐹(𝑧 + ∆𝑧) Thus, the heating experienced by a layer of air due to radiation transfer may be expressed in terms of the rate of temperature change. It is conventionally given by 𝜕𝑇 ∆𝐹 𝑧 = −𝜌𝑐𝑝 ∆𝑧 𝜕𝑡  The heating rate for a differential layer ∆𝑧 is, therefore, here we have also expressed the heating rate in terms of pressure and path-length coordinates using the hydrostatic equation dp = -ρg dz, and the definition of path length for a specific gas where q is the mixing ratio, g is the gravitational acceleration, and g/Cp is the well-known dry adiabatic lapse rate. For N layers in the atmosphere To compute the solar flux and heating rate in a clear atmosphere, we must include absorption by various absorbing gases, chiefly H2O, O3, O2, and CO2, and scattering by molecules and aerosols, as well as reflection from the surface. The solar spectrum must be divided into a number of suitably grouped subspectral intervals. In Fig. we show typical solar heating rates and net flux profiles as functions of the cosine of the solar zenith angle μ0 using the standard atmospheric profiles for H2O, O3, and other trace gases, along with a surface albedo of 0.1, as inputs of a radiative transfer model. The instantaneous solar heating rate profile is divided into two different levels to highlight the contributions from H2O and O3. The solar heating rate decreases as μ0 decreases because the incoming solar irradiance available to the atmosphere is directly proportional to μ0. Below about 10 km, the solar heating rate is primarily produced by water vapor with the heating rate ranging from 0.5 to 2 K day-1 near the surface when the contribution from aerosols is not accounted for. The solar heating rate decreases rapidly with increasing altitude in phase with the exponential decrease of water vapor and reaches a minimum at about 15 km. Above 20 km, increased solar heating is produced primarily by the absorption of ozone. When a standard aerosol profile with an optical depth of 0.15 Solar net flux decreases significantly below at the 0.5 μm wavelength is added, the solar heating rate about 10 km. increases in the lower atmosphere because of the absorption of aerosols in the visible and near infrared.  Effects of clouds The effects of clouds on solar heating and net flux profiles are investigated using typical single-layer cirrostratus (Cs) and stratus (St) clouds whose locations are shown in Fig., with the solar zenith angle of 0.5. The visible optical depths for Cs and St are 0.7 and 10, respectively, while the mean ice crystal maximum dimension and water droplet radius are 42 and 8 μm, respectively. In the case of low stratus, substantial instantaneous heating occurs at the cloud top with a value of about 22 K day-1. Because of the reflection from clouds, ozone heating also increases. This increase appears to depend on the factors associated with cloud position and optical depth. In the overcast low stratus condition, net solar flux available at the surface is only about 187 W m-2, in comparison to about 435 and 376 W m-2 in clear and cirrus cloud conditions, respectively.