Gaussian Quadrature and Its Application to Infrared Radiation

The Gaussian integration of moments is systematically discussed. It is shown that the well-known diffusivityfactor approximation is equivalent to a one-node Gaussian quadrature. The limit as the moment power approaches infinity in a one-node Gaussian quadrature produces a diffusivity factor of e1/2 5 1.648 721 3, which is very close to the value of 1.66 suggested by Elsasser. The errors due to the diffusivity-factor approximation are analyzed in a one-dimensional radiative transfer model. Generally, the results cannot be improved by using other one-node Gaussian quadrature schemes with different moments. More accurate results can be obtained by using higher-node Gaussian quadratures. It is found that the limit as the moment power approaches infinity always produces the best results. The computational advantage of the diffusivity-factor approximation is kept in the higher-node Gaussian quadratures. It is, therefore, feasible to implement the higher-node Gaussian quadratures in climate models.