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Scattering and Absorption by Aerosol and Cloud Particles. Mie Theory

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Scattering and Absorption by Aerosol and Cloud Particles. Mie Theory Lecture 6. Scattering and absorption by aerosol and cloud particles. Mie theory. Main radiation law (Beer-Bouger- Lambert law) 1. Scattering and absorption by aerosol and cloud particles. 2. Main radiation law (Beer-Bouguer-Lambert law). 3. Remote sensing applications based on measurements of the direct solar radiation. Required reading: S: 5.3-5.6; Petty: 12, 7.2, 7.4.2 Additional reading S: 4.2; 5.5 Advanced reading Bohren, C. F., and D. R. Huffman, Absorption and scattering of light by small particles. John Wiley&Sons, New York, pp. 531, 1983. 1. Scattering and absorption by aerosol and cloud particles: Mie theory. Mie theory describes the scattering and absorption of electromagnetic radiation by spherical particles through solving the Maxwell equations. NOTE: Mie theory is also called Lorenz-Mie theory or Lorenz-Mie-Debye theory. Figure 6.1 Simplified visualization of scattering of an incident EM wave by a particle. 1 ==================== Mie theory outline ============================ General outline of Mie theory: Key Assumptions: i) Particle is a sphere; ii) Particle is homogeneous (therefore it is characterized by a single refractive index m=n - ik at a given wavelength); NOTE: Mie theory requires the relative refractive index that is the refractive index of a particle divided by the refractive index of a medium. For air m is about 1, so one needs to know the refractive index of the particle (i.e., refractive index of the material of which the particle is composed). NOTE: If a particle has complex chemical composition such as some atmospheric aerosols, the effective refractive index must be calculated at a given wavelength. • Mie theory calculates the scattered electromagnetic field at all points within the particle (called internal field) and at all points of the homogeneous medium in which the particle is embedded. For all practical applications in the atmosphere, light scattering observations are carried out in the far-field zone (i.e., at the large distances from a particle). In the far-field zone (i.e., at the large distances R from a sphere), the solution of the wave equation can be obtained as s i ⎡El ⎤ exp( −ikR + ikz ) ⎡S 2 S 3 ⎤⎡El ⎤ ⎢ ⎥ = ⎢ ⎥ [6.1] s ikR ⎢S S ⎥ i ⎣⎢E r ⎦⎥ ⎣ 4 1 ⎦⎣⎢E r ⎦⎥ i i here k =2π/λ, El and Er are the parallel and perpendicular components of incident s s electrical field, and El and Er are the parallel and perpendicular components of scattered electrical field, ⎡S 2 (Θ) S 3 (Θ)⎤ ⎢ ⎥ is the amplitude scattering matrix (unitless) ⎣S 4 (Θ) S1 (Θ)⎦ 2 For spheres: S3(Θ) = S4(Θ) = 0, and thus Eq.[6.1] gives s i ⎡El ⎤ exp( −ikR + ikz) ⎡S 2 (Θ) 0 ⎤⎡El ⎤ ⎢ ⎥ = ⎢ ⎥ [6.2] s ikR ⎢ 0 S (Θ)⎥ i ⎣⎢E r ⎦⎥ ⎣ 1 ⎦⎣⎢E r ⎦⎥ Eq.[6.2] is a fundamental equation of scattered radiation by a sphere including polarization. Mie theory solution for the scattering amplitudes: ∞ 2 n + 1 S 1 (Θ ) = ∑ []a nπ n (cos Θ ) + b nτ n (cos Θ ) [6.3] n =1 n (n + 1) ∞ 2 n + 1 S 2 (Θ ) = ∑ []b n π n (cos Θ ) + a nτ n (cos Θ ) [6.4] n =1 n (n + 1) where πn and τn are Mie angular functions 1 π (cos Θ ) = P 1 (cos Θ ) [6.5] n sin( Θ ) n d τ (cos Θ ) = P 1 (cos Θ ) [6.6] n dΘ n 1 where Pn are the associated Legendre polynomials. Mie theory also gives the scattering phase matrix P(Θ) that relates the Stokes parameters {I0, Q0, U0 and V0 } of incident radiation field and the Stokes parameters {I, Q, U and V}of scattered radiation: ⎡I ⎤ ⎡I o ⎤ ⎢ ⎥ ⎢Q⎥ Q ⎢ ⎥ σ s ⎢ o ⎥ = 2 P [6.7] ⎢U ⎥ 4πr ⎢U 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣V ⎦ ⎣Vo ⎦ where ⎡P11 P12 0 0 ⎤ ⎢P P 0 0 ⎥ ⎢ 12 22 ⎥ P = [6.8] ⎢ 0 0 P33 − P34 ⎥ ⎢ ⎥ ⎣ 0 0 P34 P44 ⎦ 3 NOTE: In general, for a particle of any shape, the scattering phase matrix consists of 16 independent elements, but for a sphere this number reduces to four. For spheres: P22 = P11 and P 44 = P33 Thus for spheres, Eq.[6.7] reduces to ⎡I ⎤ ⎡ P11 P12 0 0 ⎤ ⎡I o ⎤ ⎢ ⎥ ⎢Q ⎥ ⎢P P 0 0 ⎥ Q ⎢ ⎥ σ s ⎢ 12 11 ⎥ ⎢ o ⎥ [6.9] = 2 ⎢U ⎥ 4πr ⎢ 0 0 P33 − P34 ⎥ ⎢U 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣V ⎦ ⎣ 0 0 P34 P33 ⎦ ⎣V o ⎦ and each element of the scattering phase matrix is expressed via the scattering amplitudes S1(Θ) and S2(Θ). P11(Θ) =P(Θ) is the scattering phase function of a particle (introduced in Lecture 5) From Mie theory it follows that the extinction cross-section of a particle is 4π 0 σ = Re[S(0 )] [6.10] e k 2 But for the forward direction (i.e. Θ =00) from Eqs.[6.3]-[6.4], we have ∞ 0 0 1 S1(0 ) = S2 (0 ) = ∑(2n+1)(an +bn ) 2 n=1 ¾ Efficiencies (or efficiency factors) for extinction, scattering and absorption are defined as σ σ σ Q = e Q = s Q = a [6.11] e πr 2 s πr 2 a πr 2 where πr2 is the area of a article with radius r. 4 Mie theory gives the solution for Qe, Qs and Qa in terms coefficient an and bn (i.e., 0 0 coefficients in the expressions for the scattering amplitudes S1(0 ) and S2(0 )). 2 ∞ Q ( 2 n 1) Re[ a b ] [6.12] e = 2 ∑ + n + n x n = 1 ∞ 2 2 2 Q = ( 2 n + 1)[ a + b ] [6.13] s 2 ∑ n n x n = 1 and the absorption efficiency can be calculated as Q a = Q e − Q s [6.14] ----------------------------------------END of Mie theory outline---------------------------------- Figure 6.2 Examples of Qe calculated with the Mie theory for several refractive indexes. Some highlights of Mie scattering results: • Extinction efficiency vs. size parameter x (assuming NO ABSORPTION): 4 1) small in Rayleigh limit: Q e ∝ x 2) largest Qe when particles and wavelength have similar size 3) Qe -> 2 in the geometric limit ( x → ∞ ) 4) Oscillations (see Fig.6.2) from interference of transmitted and diffracted waves 5 • Period in x of interference oscillations depends on the refractive index. Absorption reduces interference oscillations and kills ripple structure. • Scattering and absorption efficiencies vs. size parameter with ABSORPTION: As x → ∞ : Q s → 1 and , entering rays are absorbed inside particle. Smaller imaginary part of the refractive index requires larger particle to fully absorb internal rays. • Scattering phase function: forward peak height increases dramatically with x. For single particles – number of oscillations in P(Θ) increases with x. For a single spherical particle the Mie theory gives the extinction, scattering and absorption cross-sections (efficiency factors), the scattering amplitudes and phase matrix. How to calculate optical characteristics of an ensemble of spherical particles: Particle size r Refractive index, m(λ) Mie theory Scattering cross section, σs Absorption cross section, σa Extinction cross section, σe Phase function, P11(Θ) (as a function of a particle size and wavelength) Integration over size distribution N(r) Scattering coefficient, κs Absorption coefficient, κa Extinction coefficient, κe Phase function, P11(Θ) (as a function of the wavelength) 6 Integration over the particle size distribution: For a given type of particles characterized by the size distribution N(r)dr, the volume extinction, scattering and absorption coefficients (in units LENGTH-1) are determined as r2 r2 k = σ (r)N(r)dr = πr 2Q N(r)dr [6.15] e ∫∫e e r1 r1 r2 r2 k = σ (r)N(r)dr = πr 2Q N(r)dr [6.16] s ∫∫s s r1 r1 r2 r2 k = σ (r)N(r)dr = πr 2Q N(r)dr a ∫ a ∫ a [6.17] r1 r1 Similar to Eqs.[4.11 and 4.12], the optical depth of an aerosol layer (between s1 and s2) is defined as s2 s2 τ (s , s ) = k ds = Mk ds λ 1 2 ∫ e ,λ ∫ m ,e ,λ s1 s1 where km,e,λ is the mass extinction coefficient and M is the mass of partcles NOTE : Mass coefficients = volume coefficients/particle mass concentration, M The single scattering albedo gives the percentage of light which will be scattered in a single scattered event and it is defined as k s ω 0 = [6.18] k e NOTE: No absorption (conservative scattering): ω0 = 1 No scattering: ω0 = 0 Scattering phase function of particles characterized by the size distribution N(r)dr r2 P(Θ, r)σ N(r)dr ∫ s P(Θ) = r1 [6.19] r2 σ N(r)dr ∫ s r1 7 Figure 6.3 Examples of representative scattering phase functions (at a wavelength of 0.5 µm) for aerosol and clouds particles. The molecular (Rayleigh) is also shown for comparison. 8 ¾ Optical properties of the external mixture (i.e., the mixture of several types of particles) i i i ke = ∑ ke ks = ∑ks ka = ∑ka [6.20] i i i i i i where ke , ks and ka are calculated for each particle type characterized by its particle size distribution Ni(r) and a refractive index (or effective refractive index) mi. NOTE: Do not sum the single scattering albedo and scattering phase functions!!! See below Eq.[6.22-6.24] how to do it wright. ¾ How to calculate the effective optical properties of an atmospheric layer consisting of gas and aerosols (or clouds): In general, an atmospheric layer has molecules, aerosols and/or cloud particles. Thus, one needs to calculate the effective optical properties of this layer as an external mixture of the optical properties of these constituents.
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