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Two-stream approximation

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In models of radiative transfer, the two-stream approximation is a discrete ordinate approximation in which radiation propagating along only two discrete directions is considered. In other words, the two-stream approximation assumes the intensity is constant with angle in the upward hemisphere, with a different constant value in the downward hemisphere. It was first used by Arthur Schuster in 1905.cite-ref-1[1] The two ordinates are chosen such that the model captures the essence of radiative transport in light scattering atmospheres.cite-ref-meador80a-2-0[2] A practical benefit of the approach is that it reduces the computational cost of integrating the radiative transfer equation. The two-stream approximation is commonly used in parameterizations of radiative transport in global circulation models and in weather forecasting models, such as the WRF. There are a large number of applications of the two-stream approximation, including variants such as the Kubelka-Munk approximation. It is the simplest approximation that can be used to explain common observations inexplicable by single-scattering arguments, such as the brightness and color of the clear sky, the brightness of clouds, the whiteness of a glass of milk, and the darkening of sand upon wetting.cite-ref-3[3] The two-stream approximation comes in many variants, such as the Quadrature, and Hemispheric constant models.cite-ref-meador80a-2-1[2] Mathematical descriptions of the two-stream approximation are given in several books.cite-ref-4[4]cite-ref-5[5] The two-stream approximation is separate from the Eddington approximation (and its derivatives such as Delta-Eddingtoncite-ref-6[6]), which instead assumes that the intensity is linear in the cosine of the incidence angle (from +1 to -1), with no discontinuity at the horizon.cite-ref-7[7]

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