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Four-Stream Spherical Harmonic Expansion Approximation for Solar Radiative Transfer

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  • 1 Atmospheric and Oceanic Sciences Program, Princeton University, Princeton New Jersey
  • | 2 NOAA/Geophysical Fluid Dynnamics Laboratory, Princeton University, Princeton New Jersey
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Abstract

This paper presents a four-stream extension of the δ-Eddington approximation by considering the higher-order spherical harmonic expansion in radiative intensity. By using the orthogonality relation of the spherical harmonic functions, the derivation of the solution is fairly straightforward. Calculations show that the δ-four-stream spherical harmonic expansion approximation can reduce the errors in reflection, transmission, and absorption substantially in comparison with the δ-Eddington approximation. For the conservative scattering case, the error of the new model is generally less than 1% for optical thickness greater than unity except for gracing incident solar beam. For nonconservative scattering cases (single scattering albedo ω=0.9), the error is less than 5% for optical thickness greater than unity, in contrast to errors of up to 20% or more under the δ-Eddington approximation. This model can also predict the azimuthally averaged intensity to a good degree of accuracy. The computational time for this model is not as intensive as for the rigorous numerical methods, owing to the analytical form of the derived solution.

Abstract

This paper presents a four-stream extension of the δ-Eddington approximation by considering the higher-order spherical harmonic expansion in radiative intensity. By using the orthogonality relation of the spherical harmonic functions, the derivation of the solution is fairly straightforward. Calculations show that the δ-four-stream spherical harmonic expansion approximation can reduce the errors in reflection, transmission, and absorption substantially in comparison with the δ-Eddington approximation. For the conservative scattering case, the error of the new model is generally less than 1% for optical thickness greater than unity except for gracing incident solar beam. For nonconservative scattering cases (single scattering albedo ω=0.9), the error is less than 5% for optical thickness greater than unity, in contrast to errors of up to 20% or more under the δ-Eddington approximation. This model can also predict the azimuthally averaged intensity to a good degree of accuracy. The computational time for this model is not as intensive as for the rigorous numerical methods, owing to the analytical form of the derived solution.

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