The Delta-Eddington Approximation for Radiative Flux Transfer

TLDR

The study introduces the delta‑Eddington approximation, a rapid yet accurate method for computing monochromatic radiative fluxes in absorbing‑scattering atmospheres. It achieves this by combining a Dirac delta function with a two‑term approximation, assigning the forward‑peak scattering proportionally to the square of the phase‑function asymmetry factor, thereby improving accuracy for highly asymmetric phase functions compared to the standard Eddington approach. Comparisons with exact calculations show typical errors of 0–0.022, maximum 0.15, and flux errors of no more than 0.5 % on average and 2 % at worst, demonstrating that the method is both computationally fast and accurate enough for applications such as general circulation and climate modelling.

Abstract

This paper presents a rapid yet accurate method, the "delta-Eddington" approximation, for calculating monochromatic radiative fluxes in an absorbing-scattering atmosphere. By combining a Dirac delta function and a two-term approximation, it overcomes the poor accuracy of the Eddington approximation for highly asymmetric phase functions. The fraction of scattering into the truncated forward peak is taken proportional to the square of the phase function asymmetry factor, which distinguishes the delta-Eddington approximation from others of similar nature. Comparisons of delta-Eddington albedos, transnmissivities and absorptivities with more exact calculations reveal typical differences of 0–0.022 and maximum differences of 0.15 over wide ranges of optical depth, sun angle, surface albedo, single-scattering albedo and phase function asymmetry. Delta-Eddington fluxes are in error, on the average, by no more than 0.5%0, and at the maximum by no more than 2% of the incident flux. This computationally fast and accurate approximation is potentially of utility in applications such as general circulation and climate modelling.