The Two-Stream Approximation in Radiative Transfer: Including the Angle of the Incident Radiation

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Abstract
The two-stream approximation has been applied to the equation of radiative transfer to obtain two–stream models for the transfer of radiation through an optically thin plane-parallel atmosphere. The models include the dependence of the reflection and the transmission of the atmosphere on the angle of the incident radiation and on the angular dependence of the scattering phase function of the medium. The two models arise from different methods for treating the incident radiation. It is shown that the models reduce to the thin-atmosphere approximation in the limit that the optical depth of the atmosphere approaches zero. In this limit the sign of the heating caused by the presence of a scattering and absorbing layer over a reflecting surface is derived. This reveals the importance of both the zenith angle and the angular dependence of the scattering phase function. The results obtained from the two-stream models are compared with those of numerical solutions to the equation of radiative transfer. It is found that the results are adequate for layers with optical depths of a few tenths even for moderately large angles of the incident radiation.