Two-and Four-stream Combination Approximations For Computation Of Diffuse Actinic Fluxes

Abstract: In this paper, theδ-two-and four-stream combination approximations, which use a source function from the two-stream approximations and evaluate intensities in the four-stream directions, are formulated for the calculation of diffuse actinic fluxes. The accuracy and efficiency of the three computational techniques, i.e., theδ-two-stream approximations, theδ-two- and four-stream combination approximations based on various two-stream approaches, and theδ-four-stream approximation, have been investigated. The diffuse actinic fluxes are examined by considering molecular, aerosol, haze and cloud scattering over a wide range of solar zenith angles, optical depths, and surface albedos. In addition, the following conclusions are mainly achieved:(1) In general, for a conservative or non-conservative Rayleigh scattering atmosphere, the accuracy of theδ-two- and four-stream combination approximations based on various two-stream approaches is close to that of theδ-four-stream approximation and much more accurate than that of theδ-two-stream approximations.(2) For non-absorbing aerosols, when the optical depthτ1 is equal to 0.02, 2.5, and 5, the accuracy of theδ-two- and four-stream combination approximation based on the quadrature scheme is similar to that of theδ-four-stream approximation. In other cases, theδ-four-stream approximation performs better. Hence theδ-two- and four-stream combination approximation based on the quadrature scheme is better than those based on the Eddington and hemispheric constant schemes.(3) For both Cloud C1 and Haze L, the accuracy ofδ-two- and four-stream combination approximation based on the Eddington hemispheric constant scheme is slightly better than or similar to that of theδ-two-stream approximation based on the Eddington and hemispheric constant schemes. In addition, theδ-two-and four-stream combination approximation based on the quadrature scheme is better than those based on the Eddington and hemispheric constant schemes.(4) We found that for the calculation of diffuse actinic fluxes, the errors in theδ-two-stream methods were largest. For theδ-four-stream approximation, reliable results were obtained under all conditions. The accuracy of theδ-two- and four-stream combination approximations based on the quadrature scheme was close to that of theδ-four-stream approximation.In view of their accuracy and computational efficiency, theδ-two- and four-stream combination method based on the quadrature scheme takes the advantages of the speed of the two-stream approximations and the accuracy of the four-stream approximation. Hence it is well-suited to diffuse actinic flux calculations.(5) The new scheme to deal with diffusivity factor, which is more precise in comparison with the previous studies, is presented. For diffusivity factor and the previous ones are put into a radiative transfer model to calculate the atmospheric heating rate. The results show that the new scheme is generally better than previous ones, especially in the upper atmosphere with a height of more than 45 km.