An Application of the Discrete Ordinate Method to Terrestrial Radiation in Climate Models

Abstract

Thermal infrared radiation is so incorporated into the discrete ordinate method as to be usable in climate models. A source term for thermal infrared radiation contains up to the second derivative, set to be a constant, of the Planck function with respect to optical thickness, and this treatment (quadratic approximation) is indispensable for ensuring reasonable accuracy in climate models. This is because the models prognostically yield not a continuous temperature profile but a discontinuous layer-mean temperature, from which temperatures at full level and half level are diagnostically determined. Then, the temperature profile presumed from these temperatures comes to show curvature (nonlinearity) within one layer where the layer-mean temperature is minimum or maximum. Assuming that the temperature profile between adjacent full levels is linearly represented with pressure, radiative?convective equilibrium temperatures are calculated with a four-stream approximation in order to examine the accuracy of the quadratic (QD-EXT) and linear (LN-EXT) approximations of the Planck function. Their results are compared with that of an ?exact? calculation, in which the number of layers is doubled. In addition, a comparison is also made for the method of treating isothermal temperatures in the maximum and minimum temperatures with the quadratic (QD-ISO) and linear (LN-ISO) approximations. It is demonstrated that the linear approximations, LN-EXT and LN-ISO, yield prominent two-grid noises with large errors in the troposphere. The quadratic approximations, QD-EXT and QD-ISO, on the other hand, show reasonable accuracy with small errors in the troposphere. QD-EXT is found to be the most suitable method for climate models because of its accuracy and simplicity.