Multiple scaling method and its application to the calculation of radiative fluxes

New scaling formulae to calculate radiative fluxes were derived. First, we showed the multiple scaling method in which the scaling factor was introduced independently for each scattering order on the basis of the successive orders method. This formula cannot be solved fast except for the optically thin atmospheres. Then we transformed the multiple scaling method into the double scaling method which scales the ordinary radiative transfer equation by two factors.

Using discrete ordinates method with two-stream and four-stream approximations, we solved the radiative transfer equations scaled by the double-scaling and delta-M methods. Comparing the results of these two scaling methods, we found that the double scaling method improves the accuracy of radiative fluxes at large solar zenith angles, especially, in the optically thin case, and that its accuracy is comparable to that of the delta-M method in the case multiple scattering dominates. Once appropriate scaling factors are determined, the double scaling method calculates radiative fluxes as fast as the delta-M method. This method, therefore, is useful to compute radiative fluxes in general circulation models.