Scaling algorithms for the calculation of solar radiative fluxes

We derived new scaling formulae based on the method of successive orders of scattering to calculate solar radiative flux. In this report, we demonstrate a multiple scaling method, in which we introduce scaling factors for each scattering order independently. The formula of radiative transfer by the method of successive orders of scattering cannot be solved rapidly except in the case of optically thin atmospheres. Then we further derived a double scaling method, which scales the ordinary radiative transfer equation by two scaling factors. We applied the double scaling method to two-stream and four-stream approximations of the discrete ordinates method. Comparing the results of the double scaling method with those of the delta-M method, we found that the double scaling method improved the accuracy of radiative fluxes at large solar zenith angles, especially in the optically thin region, and that in the region where multiple scattering dominates, its accuracy was comparable to that of the delta-M method. Once we determined the scaling factors appropriately, the double scaling method calculated radiative fluxes as rapidly as the delta-M method in the two-stream and four-stream approximations. This method, therefore, is useful for accurate computation of solar radiative fluxes in general circulation models.