Construction of the multi-layers four-stream spherical harmonic expansion algorithm and its application to atmospheric radiative model

In order to increase the accuracy of radiative transfer calculation without increasing calculation cost, a method using the single-layer four-stream spherical harmonic expansion approximation combining with the two-stream adding algorithm has been built in this paper. Then, the four-stream spherical harmonic expansion approximation was compared with the traditional Eddington approximation and the four-stream discrete ordinate approximation by using the 48-stream discrete-ordinate approximation method as a standard reference. The results of the upward and downward radiative fluxes as well as the heating rates in the real atmosphere have been shown. For clear sky, the differences between the Eddington method, four-stream spherical harmonic expansion approximation, four-stream discrete-ordinate approximation, and the standard results are all less than 0.3 K/d for heating rates, and less than 1% and 0.6% for the upward and downward radiative fluxes, respectively. It shows that the three methods have no large difference for heating rate calculations. The two four-stream approximations are a little more accurate than the Eddington method for flux calculation in clear sky. For cloudy sky, the differences of heating rates between the two four-stream methods and the standard results are all less than 1% at the top of the cloud, whereas these differences between the Eddington method and the standard results are large than 5%. Therefore, the four-stream spherical harmonic expansion approximation built in this paper can be applied to the radiative transfer model to improve the accuracies of the whole radiation calculation for clear sky and the heating rate calculation for cloudy sky without increasing large costs.