A reliable and efficient two-stream algorithm for spherical radiative transfer: Documentation of accuracy in realistic layered media

Abstract

We present a fast and well documented two-stream algorithm for radiative transfer and particle transport in vertically inhomogeneous, layered media. The physical processes considered are internal production (emission), scattering, absorption, and Lambertian reflection at the lower boundary. The medium may be forced by internal sources as well as by parallel or uniform incidence at the top boundary. This two-stream algorithm is based on a general purpose multi-stream discrete ordinate algorithm released previously. It incorporates all the advanced features of this well-tested and unconditionally stable algorithm, and includes two new features: (i) corrections for spherical geometry, and (ii) an efficient treatment of internal sources that vary rapidly with depth. It may be used to compute fluxes, flux divergences and mean intensities (actinic fluxes) at any depth in the medium. We have used the numerical code to investigate the accuracy of the two-stream approximation in vertically inhomogeneous media. In particular, computations of photodissociation and warming/cooling rates and surface fluxes of ultraviolet and visible radiation for clear, cloudy and aerosol-loaded atmospheres are presented and compared with results from multi-stream computations. The O3 +hv → O(1D) + O2 and O3 +hv → O(3P) + O2 photodissociation rates were considered for solar zenith angles between 0.0–70.0° and surface albedos in the range 0.0–1.0. For small and moderate values of the solar zenith angle and the surface albedo the error made by the two-stream approximation is generally smaller, <10%, than the combined uncertainty in cross sections and quantum yields. Surface ultraviolet and visible fluxes were calculated for the same range of solar zenith angles and surface albedos as the photodissociation rates. It was found that surface ultraviolet and visible fluxes may be calculated by the two-stream approximation with 10% error or less for solar zenith angles less than 60.0° and surface albedos less than 0.5. For large solar zenith angles and/or large surface albedos, conditions typical at high latitudes, the error made by the two-stream approximation may become appreciable, i.e. 20% or more for the photodissociation rates in the lower stratosphere and for ultraviolet and visible surface fluxes for large surface albedos. The two-stream approximation agrees well with multi-stream results for computation of warming/cooling rates except for layers containing cloud and aerosol particles where errors up to 10% may occur. The numerical code provides a fast, well-tested and robust two-stream radiative transfer program that can be used as a ‘software tool’ by aeronomers, atmospheric physicists and chemists, climate modellers, meteorologists, photobiologists and others concerned with radiation or particle transport problems. Copies of the FORTRAN77 program are available to interested users.