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Based on the unified formulation, the so-called hemispherical harmonics method with four components (HSHM4) is derived following the general procedure described in the first paper of this series. The numerical results of this method, including the flux reflections, transmissions, and absorptions for different optical depths, incident beam angles, and single-scattering albedos and the emissions for different optical depths and single-scattering albedos, are reported, in comparison with the results of DISORT with 128 streams as a benchmark. The application of the method to the radiation scheme for climate models is illustrated with the numerical results, including the heating-rate profiles for the shortwave and longwave radiation, radiative fluxes at the top and the bottom boundaries of the atmosphere, and the actinic flux profiles, for typical model atmospheres of Earth and different clear or cloudy conditions. The comparison of results with those of the discrete ordinate methods (DOMs) and the spherical harmonic method (SHM) with four components shows that the HSHM4 has comparable accuracy with the four-stream DOM with double-Gaussian quadrature and is more accurate than the four-stream SHM and the four-stream DOM with full-range Gaussian quadrature in general, especially in the cases of longwave radiation when the multiple-scattering effect should be accounted for. The HSHM4 can be deployed consistently to all of the bands without artificial division between the longwave and shortwave. The development and validation of the HSHM4 exhibit the usefulness of the unified formulation in the study of atmospheric radiative transfer.

journal of the atmospheric sciences

A Unified Formulation of Radiative Transfer in Plane-Parallel Atmospheres Based on General Decomposition of Radiance. Part II: An Exemplifying Application to the Hemispherical Harmonics Method with Four Components