Analysis of Scattering of Radiation in a Plane-Parallel Atmosphere Stephanie M. Carney ES 299r May 23, 27
TABLE OF CONTENTS. INTRODUCTION... 2. DEFINITION OF PHYSICAL QUANTITIES... 3. DERIVATION OF EQUATION OF TRANSFER... 6 4. APPROXIMATION OF EQUATION OF TRANSFER... 8 4. Review of methods used by others... 8 4.2 Approximations that we apply... 9 5. RESULTS... 6. CONCLUSIONS... 4 7. REFERENCES... 4
. INTRODUCTION When beams of radiation from the sun enter Earth s atmosphere, they are scattered. Scattering is a physical process by which particles in the path of radiation continuously extract and reradiate energy from the beam in all directions (Liou, 22). Most of the sun s light that reaches our eyes comes from this process. Scattering causes changes in radiation s intensity, or irradiance, which is the radiant flux incident upon a unit area of a surface. We are interested in the change in irradiance due to scattering in Earth s atmosphere. This paper is an extension of a project completed in a previous class with emphasis on the numerical methods involved in the solution. Here we focus on the physical significance of the equation. We then solve the equation of transfer using the discrete ordinate method. 2. DEFINITION OF PHYSICAL QUANTITIES In order to analyze radiative transfer in Earth s atmosphere, we need to derive a fundamental equation that governs the radiation field in a medium that absorbs, emits, and scatters radiation. Before we can derive the equation that governs radiative transfer in a scattering atmosphere, we need to define and understand the physical quantities that are involved. This section introduces the concepts that are necessary to form the radiative equation of transfer in a scattering atmosphere. When we study the scattering of radiation, we consider energy that is confined to an element of solid angle. This means that photons are admitted only from a pencil of radiation about a small angle (the solid angle) from a given direction, as shown in Figure, where the solid angle is denoted by ω. We are interested in the intensity of this pencil of radiation defined by ω. The solid angle is the ratio of the area σ of a surface of a sphere that has been intercepted to the radius r 2, and it is expressed in steradians (sr). A steradian is the SI unit of solid angle that, having its vertex in the center of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere. This is analogous to the way in which radians describe angles in a plane. There are 4π steradians in a sphere (AMS, 27). σ ω = () 2 r
σ ω r Figure. Demonstration of a solid angle ω that helps define the pencil of radiation, where r is the distance to the surface of the sphere and σ is a section of area of the sphere. (Source: Adapted from Wofsy, 26) For radiation analysis, we need to consider a differential elemental solid angle, as shown in Figure 2, where a portion of a sphere with central point O is constructed. In this figure, θ is the zenith angle, and φ is the azimuthal angle. A line through point O intersects an arbitrary surface of the sphere. The pencil of radiation with differential solid angle dω is shown through center of the sphere that intersects an arbitrary surface on the sphere. From the Figure, it is clear that the differential area in polar coordinates given by d σ = (rdθ)(r sin θd. () The differential solid angle is therefore dω = dσ r 2 = sin θdθdϕ (2) 2
dσ dω Figure 2. Depiction of differential solid angle dω. A pencil of radiation through element dσ is also shown in directions confined to solid angle dω. (Source: Liou, 22) In order to describe Earth s radiation field, we must specify how much energy is traveling in each direction. The quantity describing the radiation field in this manner is called the specific intensity, or irradiance, I, (measured in Wm -2 sr - / ) at frequency. In a scattering atmosphere, the scattering of radiation occurs in all directions. The irradiance is isotropic if it is independent of direction. We next consider the differential amount of radiant energy de in a specified frequency interval (, + d), which crosses an element of area dσ and is confined to the solid angle dω during time dt, as shown in Figure 3. dω φ dσ Figure 3. Depiction of energy with solid angle dω transferred across area dσ 3
Here φ is the angle the pencil of radiation forms with the normal to dσ. We can then express de in terms of I in the following way: de = I cosϕddσdωdt (3) When the pencil of radiation travels through a medium, it will be weakened by the interaction (Chandrasekhar, 96). A depiction of this is given in Figure 4, where ds is a short distance through the medium in the direction of propagation. ds I I + di Figure 4. Depiction of the change in irradiance as the beam of radiation passes through a medium. The attenuation of the light is referred to as extinction. There are two methods of extinction for a stream of light in the direction of propagation. The first method is through absorption. When the pencil of radiation travels through the medium, a portion of it will be absorbed in the sense that it represents the transformation of radiation into other forms of energy (Chandraasekhar, 96). The second method is through the scattering of the intensity into a different pencil of radiation. Due to the law of conservation of energy, this radiation is not lost; it is redirected into another pencil of radiation. It only represents a loss of radiation in the pencil of radiation that we are considering in its direction of propagation. Therefore, both scattering and absorption remove energy from the beam of light (in the direction of propagation) that is traveling through a medium. After traveling a distance ds through the medium, the specific intensity will become I + di, where di can be expressed in terms ds as κ ρi ds (4) where ρ is the density of the medium. Here we define κ as the mass absorption coefficient for radiation of a frequency. It describes how much specific intensity is lost in the direction of propagation. 4
We must also specify the angular distribution of the radiation being scattered. For this reason, we specify a phase function p( θ, ϕ, θ', ϕ' ), that is dependent upon the angle θ at which the energy is being scattered to the direction of incidence. The angle is known as the incident angle. If the beam of radiation entered the atmosphere vertically, the scattered intensity would be independent of the direction of scattering. Expression 5 gives the rate at which energy is being scattered into a medium with solid angle dω and in a direction given by θ and φ. where dm describes the mass of the element, dω' κ I p( θ, ϕ, θ', ϕ') dmddω (5) 4π dm = ρcosϕdσdω. (6) Therefore, the rate of loss of energy from the pencil of radiation entering the atmosphere due to isotropic scattering is given by dω' κ I dmddω p( θ, ϕ, θ', ϕ'). (7) 4π dω' We refer to p ( θ, ϕ, θ', ϕ') = ϖ. This represents the single scattering albedo, which 4π is the fraction of light from the beam of radiation that is lost to scattering. The value (- ϖ ) represents the fraction of lost intensity that has been absorbed by the radiation field, i.e., converted to other forms of energy, due to the law of conservation of energy. Next we must consider the emission coefficient, because the intensity may be strengthened by emission from the medium. In addition, there will be a contribution to the emission coefficient from scattering that occurs from every other direction into the pencil of radiation that we are considering. In the frequency interval (, + d), the emission coefficient j is defined in such a way that j dmdωddt (8) describes the amount of energy a mass dm emits in an area confined to solid angle dω in the frequency interval (, + d). The increase in irradiance due to both emission and scattering is given by j ρds. (9) The contribution to the intensity from scattering from θ, φ directions into the current pencil of radiation is given by the following rate: 5
dθ'dϕ' κdmd dωip( θ, ϕ, θ', ϕ')i ( θ', ϕ'). () 4π It follows from Expression that the contribution to the emission coefficient from scattering is given by π 2π s j = κ p( θ, ϕ, θ', ϕ')i ( θ', ϕ')dθ'dϕ' () 4π The consideration of emission and absorption leads to the definition of the source function, J, which is a very important element of radiative transfer. The source function is defined as the ratio of the absorption coefficient, κ, to the emission coefficient j. According to Equation, in a scattering atmosphere, the source function is equal to π 2π J = p( θ, ϕ, θ', ϕ')i ( θ', ϕ')dθ'dϕ' (2) 4π There are other terms that are useful when analyzing radiative transfer in a scattering atmosphere. The mean intensity is simply average of specific intensity over the sphere that we are considering, mainly the Earth: 2π π I = dϕ sin θdθi (3) 4π 3. DERIVATION OF EQUATION OF TRANSFER Having defined the physical quantities involved in the analysis of radiative transfer, we are now ready to derive the equation of transfer. This is the fundamental equation that determines the change in intensity of a beam of radiation through a medium characterized by the absorption and scattering coefficients. Most analysis of radiation in an atmosphere that scatters, absorbs, and emits radiation requires the definition of an equation of transfer. As in the last section, we consider a beam of radiation traveling a distance ds through the medium, over the frequency interval (, + d). The equation of transfer, therefore, is known as the change in energy from the excess in emission over absorption in the frequency interval. Counting up the gains and losses given by in Expressions 4 and 9 from scattering and emission, we have 6
di ds = κρi + jρ (4) We can also rewrite Equation 4 in terms of the source function J. Knowing that the source function is the ratio of the emission coefficient to the absorption coefficient, Equation 4 becomes di κ ρds = I J (5) We need to consider how to treat the change in density and mass absorption coefficient. This can be accounted for by considering optical depth, τ. The change in irradiance can be thought of as a function of the optical depth, which measures the depletion that a beam of radiation would experience passing through the atmosphere (Wallace & Hobbs, 26). Considering this, the change in optical depth can be noted as d τ = κρdsμ (6) where μ is the cosine of the incident angle θ. It is customary to use μ = cos θ as an independent variable instead of θ. Since θ ranges from to π, μ will range from - to. Negative values of μ correspond to downward propagating radiation, and positive values of μ correspond to upward propagating radiation. We also adopt the notation I( τ, μ, for the values of upward radiation and I( τ, μ, for downward radiation. Using Equation 6, Equation 5 can be rewritten as di μ dτ = I J (7) The equation of transfer, therefore, is known as the change in energy from the excess in emission over absorption in the frequency interval. Since the source function is dependent upon the radiation at a point, Equation 5 contains an integral and is therefore known as an integro-differential equation. This fact is one reason why radiative transfer is so computationally difficult. The complexity of this equation requires that approximations be used to find a solution for the irradiance. From this point on, Chandrasekhar, 96 and many other researchers drop the subscript to simplify notation, and I follow suit. We need to consider the boundary conditions of the equation of transfer before we consider how to approximate the solution. 7
The most general boundary conditions we use to solve the equation of transfer are I( τ =, μ, = I ( μ, (8a) I( τ = τ*, +μ, = I *( μ, (8b) Here we note that τ = τ* refers to the bottom of the cloud or other scattering medium, and τ = refers to the top. Therefore, the first boundary condition states that downward radiation at the top of the atmosphere is equal to the intensity at the top of the atmosphere, and the bottom boundary condition refers to upward radiation at the bottom of the atmosphere. 4. APPROXIMATION OF EQUATION OF TRANSFER 4. Review of methods used by others Over the years, many techniques have been developed in attempt to solve the problem of understanding the dynamics of radiative transfer in systems containing clouds and also aerosols. There is much complexity introduced by the heterogeneous composition of real-world atmospheres, which results in mathematical systems that are considered intractable by classical methods (Liou, 973). For these reasons, numerical methods have been developed by many researchers to approximate the solution. Both the atmosphere and the ocean may be regarded as thin, stratified layers. Thermal and optical properties vary over short vertical scales (.5 km), but variations are usually over much larger scales (, km) in the horizontal direction. Also, the atmosphere is considered to be thin because its depth is much smaller than the radius of the Earth. The effect of curvature on a photon is usually small. Many of the numerical techniques designed have been crafted with the purpose of discovering solutions to the problem of radiative transfer under very specific conditions. A problem with these techniques and others like them are that they lack the generality to be universally applied. In response to these deficiencies, researchers developed a more general system based on the work of Chandrasekhar (96) in Discrete Ordinates. The Discrete Ordinate Method is superior to those mentioned before in that is designed in such a way as to factor in the physical quantities described in the second section of this paper. Advances in the development of numerical techniques to the solutions of differential equations and increased availability of the computation power with which to execute them has led researchers to resurrect the Chandrasekhar method and create well documented, versatile code for researchers to use (Stamnes, et al 988). In order to demonstrate the veracity of the results gleaned from the discrete ordinate method, comparisons have been made between discrete ordinate generated results and 8
results of other methods which are known to be correct. Specifically, the discrete ordinate method was compared with the doubling method developed by van De Hulst (963). The results generated by the discrete ordinate method resulted in negligible differences for most optical depths, on the order of.. 4.2 Approximations that we apply As stated earlier, in order to solve the integro-differential equation, we need to make approximations for computational simplicity. The first approximation that is made by Chandrasekhar, 96 and most other researchers is the plane parallel approximation. We define our atmosphere as plane parallel in a finite atmosphere, which is an approximation that depicts the atmosphere as one-dimensional and bounded at the top and bottom by horizontal plane surfaces, as shown in Figure 5. In this approximation, variations in intensity and atmospheric parameters are permitted only in the vertical direction (Liou, 22). Linear distances are measured normal to the plane of stratification. Figure 5 also depicts the upward and downward facing intensities for a finite plane-parallel atmosphere that is bounded according to the general conditions in Equation 8. I( τ =, μ, Top I( τ =, μ, τ = I( τ, μ, τ I( τ, μ, Bottom I( τ = τ*, μ, τ = τ* I( τ = τ*, μ, Figure 5. Depiction of a finite plane-parallel atmosphere and the associated upward and downward intensities at different levels of optical depth. (Adapted from Liou, 22) 9
In a plane parallel atmosphere, Equation 7 can be expressed as the change in intensity with optical depth, τ, as in the following equation: 2 di ( τ, μ, μ = I( τ, μ, dτ 4π π p( μ, ϕ, μ', ϕ')i ( τ, μ', ϕ')dμ' dϕ' (9) which is the fundamental equation of transfer for a plane parallel atmosphere. We can now define more specific boundary conditions for the plane parallel atmosphere. Chandrasekhar, 96 states that the standard problem for the equation of transfer is to solve it with the following specific boundary conditions for diffuse solar radiation: I(,-μ)= This means that at the top of the atmosphere, no downward radiation is scattered, i.e., no downward scattering occurs at the top of the atmosphere because there is no place for incoming radiation to be scattered. τ * / μ I (τ *, μ )= AI + AI / 4πe, where τ * is the bottom of the atmosphere. This means that the upward irradiance at the bottom of the atmosphere is equal to the downward irradiance multiplied by the albedo of the ground. Therefore, the ground reflects a portion of all incoming radiation, depending upon the albedo. Many approximations involve the confinement of scattering to a finite number of directions, including Gaussian Quadrature, a method employed by Chandrasekhar, 96. This method of integration allows greater accuracy because one is given freedom to choose weighting coefficients and the abscissas at which the function is evaluated. The complexity of the solution increases as the number of directions in the approximation increases. Liou (973) found that it is not possible to determine an analytic solution for more than four directions without complicated numerical methods, due to the complexity of the simultaneous differential equations.
5. RESULTS Chandrasekhar, 96 first followed the example set by Schuster (95) and Schwarzschild (96) by solving the simplest equation of transfer that is independent of azimuth angle: di ( τ, μ) μ = I( τ, μ) dτ 2 I ( τ, μ')dμ' (2) The next step is to break the intensity into 2 streams, upward and downward, I+ and I-, respectively, where di + 2 dτ = I + 2 ( I + I ) + (2a) di 2 dτ = I 2 ( I + I ) + (2b) It follows from this simple approach that we can divide the radiation field into 2n streams in the direction μ i where i=(+/-, 2, 3, +/-n), and we can form a new system of equations from Equation 9. di ( τ, μi ) μi = I( τ, μi ) a ji( τ, μ j) dτ 2 j (22) where the a j s are the weights for the quadrature formula. We can use Legendre polynomials to represent the weights in the following way: Pm ( μ) a j = dμ (23) P' ( μ ) μ μ m j j In addition, the phase function can be expanded as a series of Legendre polynomials, in the form N p ( θ, ϕ, θ', ϕ') = ϖl Pl ( μ j). (24) Each ϖ l is a constant. When solving radiative transfer problems numerically, we only consider a finite number of terms for the series. l=
From here, Chandrasekhar, 96 computes the equation of transfer according to the following sums: μ i dii ( τ, μi ) dτ = I ( τ, μ ) i i 2 N l= ϖ P ( μ ) l l i j a I P ( μ j j l j ). (25) Here Equation 25 is a system of N linear equations. We can now use a matrix differential equation of the following form: di = AI dτ Where I is a vector and A is a matrix with the following structure: (26) N ϖ Aij = a j Pl ( μi )Pl ( μ j). (27) μ j 2 2 l= In order to solve Equation 26, we look for solutions of the following form: kτ I = I e (28) We can now determine the eigenvalues and eigenvectors A. Because of the special structure of the matrix A in Equation 27, we know that the eigenvalues will be real values that occur in pairs of ±k. This was solved using the MATLAB command eig. Hence, the general solution can be written as N j= k jτ, j e I = I (29) Using the specific boundary conditions that we defined in Section 4.2, I(,-μ)=, (3a) I (τ * τ * / μ, μ )= AI + AI / 4π (3b) e solutions were found for various physical parameters. Figure 6 represents the solution of Equation 29 with τ ranging from to (the cloud has an optical depth of ), N = 4 and a single scattering albedo of.9. 2
Figure 6. Intensity as function of τ. τ= represents the top of the atmosphere, and τ= represents the bottom. The single scattering albedo is.9. Figure 7 represents the solution of Equation 29 with τ ranging from to (the cloud has an optical depth of ), N = 4 and a single scattering albedo of.7. Figure 7. Intensity as function of τ. τ= represents the top of the atmosphere, and τ= represents the bottom. The single scattering albedo is.7. 3
As can be seen in Figures 6 and 7, the intensity follows the boundary conditions specified by Equation 3. The downward intensity is equal to at the top of the atmosphere. In Figure 6, the single scattering albedo ϖ. =.9, which is a common value for the atmosphere. As you can see, the scattered radiation, is higher at the bottom of the atmosphere. In Figure 7, the scattering albedo ϖ. =.7, which implies that a small amount of radiation is scattered. As a result, the emission coefficient is much smaller and the specific intensity at the bottom of the atmosphere is much smaller. 6. CONCLUSIONS Many papers in the field of radiative transfer are focused upon the numerical methods of solving the equation of transfer. This paper describes the physical quantities involved in radiative transfer before beginning to derive the equation of transfer in an attempt to understand the meaning behind the equation. As the equation of radiative transfer is a very complicated problem with many variations, there are many different studies related to the topic. This paper uses the discrete ordinate method that replaced the integral in the equation with a discrete Gaussian quadrature. Using matrix methods, the solutions and eigenvalues were obtained using MATLAB. Since there are many ways that have been used to solve this problem for various scenarios, there any many extensions to this project that could be made. For example, a contribution to the intensity from the direct beam of radiation with no scattering could be taken into account in the equation of transfer. Since this paper is focused on the physical meaning behind the equation, there are many ways to change the parameters depending upon the specific problem. 7. REFERENCES. American Meteorological Society, Glossary of Meteorology, Second Edition, 27, http://amsglossary.allenpress.com/glossary/. 2. Chandrasekhar, S., Radiative Transfer, Dover Publications, Inc., New York, New York, 96. 3. Liou, K.N., An Introduction to Atmospheric Radiation, Second Edition, Academic Press, San Diego, California, 22. 4. Liou, K.N., A Numerical Experiment on Chandrasekhar s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres, Journal of the Atmospheric Sciences, Vol. 3, pp. 33-327, October 973. 5. Schuster, A., An Introduction to the Theory of Optics Astrophysical Journal, Vol. 2, p. 382, May 95. 4
6. Schwarzschild, K., Göttinger Nachrichten p. 4 (96). 7. Stamnes, K., Tsay, S.C., Wiscombe, W., and Jayaweera, K., Numerically Stable Algorithm for Discrete-Ordinate-Method Radiative Transfer in Multiple Scattering and Emitting Layered Media, Applied Optics, Vol. 27, No. 2, pp 252-259, June 5, 988. 8. van De Hulst, H.C., A New Look at Multiple Scattering, Tech Report, Goddard Institute for Space Studies, NASA, New York, 8 pps, 963. 9. Wallace, J.M., Hobbs, P.V., Atmospheric Science: An Introductory Survey, Second Edition, Academic Press, 26, pps 6-44.. Weinman, J.A., and Guetter, P.J., Penetration of Solar Irradiances through the Atmosphere and Plant Canopies, Journal of Applied Meteorology, Vol., pp. 36-4.. Wofsy, S., Atmospheric Chemistry and Physics, Lecture 7, Radiative Transfer, October, 26. 5