A Numerical Experiment on Chandrasekhar's Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres

The discrete-ordinate method for radiative transfer introduced originally by Chandrasekhar has been theoretically developed and numerically verified for use in solving the transfer of both solar and thermal infrared radiation through cloudy and hazy atmospheres. This method differs from other radiative transfer approaches in the sense that the solution of the transfer equation can be explicitly derived by employing a finite set of discrete-streams representing the emergent angles in the integral term. Hence such a method is practical for deriving a simplified but reliable radiative transfer approximation for meteorological applications involving clouds and aerosols. Comprehensive comparisons with other rigorous means are carried out for the transmitted and reflected intensity and flux associated with isotropic, Rayleigh and anisotropic scattering. The comparisons reveal that close agreement of radiation computations can be achieved by using discrete-streams of 16. For flux calculations, it was found that values obtained by discrete-streams of 4 yield an accuracy of about 1% for typical phase functions of clouds and haze in either solar or IR radiation, whereas the method of discrete-streams of 2 introduces errors on the order of 3–10%. The solutions for these two simplified cases can be expressed analytically. Applications have been made by employing wavelengths of 0.7, 1.5 and 10 µm to denote the transfer of solar and thermal IR irradiance through cloudy and hazy atmospheres. For solar radiation, the reflection (albedo), transmission. and absorption are obtained as functions of the zenith angle and optical thickness. It is shown that a single-scattering albedo of 0.99 produces absorption of about 30% for a cloud with an optical thickness of 20, and subsequently decreases the albedo by about 20%. The dependence of the transmitted and reflected infrared radiation on the temperatures of particulate layers and ground is illustrated for a number of thicknesses. A cloud at a temperature of −10C would normally reduce the outgoing radiation by about 40%. Interference effects due to thin haze appear to be unimportant. In addition, the net flux and the heating and cooling rates within cloud layers are presented for wavelengths of 1.5 and 10 µm. For a cloud 1 km in thickness, the solar beating takes place near the upper portion at a rate of about 0.3C hr−1 µm−1 at normal incidence, while it experiences base-warming and top-cooling at a comparable rate resulting from the thermal IR radiation. Although monochromatic wavelengths aye used in this study, the method can be extended to include the entire solar and infrared spectrum.