Adding method of delta-four-stream spherical harmonic expansion approximation for infrared radiative transfer parameterization

The adding method, which could calculate the infrared radiative transfer (IRT) in inhomogeneous atmosphere with multiple layers, has been applied to δ-four-stream discrete-ordinates method (DOM). This scheme is referred as δ-4DDA. However, there is a lack of application for adding method of δ-four-stream spherical harmonic expansion approximation (SHM) to solve infrared radiative transfer through multiple layers. In this paper, the adding method for δ-four-stream SHM (δ-4SDA) will be obtained and the accuracy of it will be evaluated as well. The result of δ-4SDA in an idealized medium with homogeneous optical property is significantly more accurate than that of the adding method for δ-two-stream DOM (δ-2DDA). The relative errors of δ-2DDA can be over 15% in thin optical depths for downward emissivity, while errors of δ-4SDA are bounded by 2%. However, the result of δ-4SDA is slightly less accurate than that of δ-4DDA. In a radiation model with realistic atmospheric profile considering gaseous transmission, the accuracy for heating rate of δ-4SDA is significantly superior than that of δ-2DDA, especially for the cloudy sky. The accuracy for heating rate of δ-4SDA is slightly less accurate than that of δ-4DDA under water cloud conditions, while it is superior than that of δ-4DDA in ice cloud cases. Beside, the computational efficiency of δ-4SDA is higher than that of δ-4DDA.