TY - JOUR
T1 - Four-stream spherical harmonic expansion approximation for solar radiative transfer
AU - Li, J.
AU - Ramaswamy, V.
PY - 1996/4/15
Y1 - 1996/4/15
N2 - 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 thicknesses greater than unity except for grazing incident solar beam. For nonconservative scattering cases (single scattering albedo ω = 0.9), the error is less than 5% for optical thicknesses 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.
AB - 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 thicknesses greater than unity except for grazing incident solar beam. For nonconservative scattering cases (single scattering albedo ω = 0.9), the error is less than 5% for optical thicknesses 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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U2 - 10.1175/1520-0469(1996)053<1174:FSSHEA>2.0.CO;2
DO - 10.1175/1520-0469(1996)053<1174:FSSHEA>2.0.CO;2
M3 - Article
AN - SCOPUS:0030466580
SN - 0022-4928
VL - 53
SP - 1174
EP - 1186
JO - Journal of the Atmospheric Sciences
JF - Journal of the Atmospheric Sciences
IS - 8
ER -