Analytical Models of Exoplanetary Atmospheres. IV. Improved Two-stream Radiative Transfer for the Treatment of Aerosols

We present a novel generalization of the two-stream method of radiative transfer, which allows for the accurate treatment of radiative transfer in the presence of strong infrared scattering by aerosols. We prove that this generalization involves only a simple modification of the coupling coefficients and transmission functions in the hemispheric two-stream method. This modification originates from allowing the ratio of the first Eddington coefficients to depart from unity. At the heart of the method is the fact that this ratio may be computed once and for all over the entire range of values of the single-scattering albedo and scattering asymmetry factor. We benchmark our improved two-stream method by calculating the fraction of flux reflected by a single atmospheric layer (the reflectivity) and comparing these calculations to those performed using a 32-stream discrete-ordinates method. We further compare our improved two-stream method to the two-stream source function (16 streams) and delta-Eddington methods, demonstrating that it is often more accurate at the order-of-magnitude level. Finally, we illustrate its accuracy using a toy model of the early Martian atmosphere hosting a cloud layer composed of carbon dioxide ice particles. The simplicity of implementation and accuracy of our improved two-stream method renders it suitable for implementation in three-dimensional general circulation models. In other words, our improved two-stream method has the ease of implementation of a standard two-stream method, but the accuracy of a 32-stream method.