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<title>Two-stream approximation</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Two-stream approximation</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In models of radiative transfer, the <b>two-stream approximation</b> 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 <a href="Arthur_Schuster" title="Arthur Schuster">Arthur Schuster</a> in 1905.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The two ordinates are chosen such that the model captures the essence of <a href="Radiative_transfer" title="Radiative transfer">radiative transport</a> in light scattering atmospheres.<sup id="cite_ref-meador80a_2-0" class="reference"><a href="#cite_note-meador80a-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 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 <a href="Global_climate_model" class="mw-redirect" title="Global climate model">global circulation models</a> and in <a href="Numerical_weather_prediction" title="Numerical weather prediction">weather forecasting models</a>, such as the <a href="Weather_Research_and_Forecasting_model" class="mw-redirect" title="Weather Research and Forecasting model">WRF</a>. There are a large number of applications of the two-stream approximation, including variants such as the <a href="Kubelka-Munk_approximation" class="mw-redirect" title="Kubelka-Munk approximation">Kubelka-Munk approximation</a>. 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.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The two-stream approximation comes in many variants, such as the Quadrature, and Hemispheric constant models.<sup id="cite_ref-meador80a_2-1" class="reference"><a href="#cite_note-meador80a-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Mathematical descriptions of the two-stream approximation are given in several books.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The two-stream approximation is separate from the <a href="Radiative_transfer#The_Eddington_approximation" title="Radiative transfer">Eddington approximation</a> (and its derivatives such as Delta-Eddington<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>), 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.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="List_of_atmospheric_radiative_transfer_codes" class="mw-redirect" title="List of atmospheric radiative transfer codes">List of atmospheric radiative transfer codes</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFLiou2002" class="citation book cs1">Liou, K. N. (2002-05-09). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mQ1DiDpX34UC&q=schuster+two+stream+climate+model"><i>An Introduction to Atmospheric Radiation</i></a>. Elsevier. p. 106. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080491677</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-10-22</span></span>.</cite></span>
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<li id="cite_note-meador80a-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-meador80a_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-meador80a_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">W.E. Meador and W.R. Weaver, 1980, Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement, 37, Journal of the Atmospheric Sciences, 630–643
<a rel="nofollow" class="external free" href="http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281980%29037%3C0630%3ATSATRT%3E2.0.CO%3B2">http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281980%29037%3C0630%3ATSATRT%3E2.0.CO%3B2</a></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Bohren, Craig F., 1987, Multiple scattering of light and some of its observable consequences, American Journal of Physics, 55, 524-533.</span>
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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFG._E._Thomas_and_K._Stamnes1999" class="citation book cs1">G. E. Thomas and K. Stamnes (1999). <i>Radiative Transfer in the Atmosphere and Ocean</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-40124-0</bdi>.</cite></span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrant_W._Petty2006" class="citation book cs1">Grant W. Petty (2006). <i>A First Course In Atmospheric Radiation (2nd Ed.)</i>. Sundog Publishing, Madison, Wisconsin. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9729033-0-5</bdi>.</cite></span>
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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFJosephWiscombeWeinman1976" class="citation journal cs1">Joseph, J. H.; Wiscombe, W. J.; Weinman, J. A. (December 1976). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%281976%29033%3C2452%3Atdeafr%3E2.0.co%3B2">"The Delta-Eddington Approximation for Radiative Flux Transfer"</a>. <i>Journal of the Atmospheric Sciences</i>. <b>33</b> (12): <span class="nowrap">2452–</span>2459. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976JAtS...33.2452J">1976JAtS...33.2452J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%281976%29033%3C2452%3Atdeafr%3E2.0.co%3B2">10.1175/1520-0469(1976)033<2452:tdeafr>2.0.co;2</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-4928">0022-4928</a>.</cite></span>
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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFShettleWeinman1970" class="citation journal cs1">Shettle, E. P.; Weinman, J. A. (October 1970). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%281970%29027%3C1048%3Attosit%3E2.0.co%3B2">"The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation"</a>. <i>Journal of the Atmospheric Sciences</i>. <b>27</b> (7): <span class="nowrap">1048–</span>1055. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970JAtS...27.1048S">1970JAtS...27.1048S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%281970%29027%3C1048%3Attosit%3E2.0.co%3B2">10.1175/1520-0469(1970)027<1048:ttosit>2.0.co;2</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-4928">0022-4928</a>.</cite></span>
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