Doubling-Adding Method for Delta-Four-Stream Spherical Harmonic Expansion Approximation in Radiative Transfer Parameterization

Though the single-layer solutions have been found for the -four-stream spherical harmonic expansion method (SHM) in radiative transfer, there is lack of a corresponding doubling-adding method (4SDA), which enables the calculation of radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The doubling-adding method is based on Chandrasekhar's invariance principle, which was originally developed for discrete ordinates approximation. It is shown that the invariance principle can also be applied to SHM and -four-stream spherical harmonic expansion doubling-adding method (-4SDA) is proposed in this paper. The -4SDA method has been systematically compared to the -Eddington doubling-adding method (-2SDA), the -two-stream discrete ordinates doubling-adding method (-2DDA), and -four-stream discrete ordinates doubling-adding method (-4DDA). By applying -4SDA to a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of -4SDA is superior to -2SDA or -2DDA, especially for the cloudy/aerosol conditions. It is shown that the relative errors of -4SDA are generally less than 1% in both heating rate and flux, while the relative errors of both -2SDA and -2DDA can be over 6%. Though -4DDA is slightly more accurate than -4SDA in heating rates, both of them are accurate enough to obtain the cloud-top solar heating. Here -4SDA is superior to -4DDA in computational efficiency. It is found that the error of aerosol radiative forcing can be up to 3 W m(-2) by using -2SDA at the top of the atmosphere (TOA); such error is substantially reduced by applying -4SDA. In view of the overall accuracy and computational efficiency, -4SDA is suitable for application in climate models.