A novel methodology for radiative transfer in a planetary atmosphere

Abstract

A novel methodology for evaluating the field of anisotropically scattered radiation within a homogeneous slab atmosphere of arbitrary optical thickness is provided. It departs from the traditional radiative transfer approach in first considering that the atmosphere is illuminated by an isotropic light source. From the solution of this problem, it subsequently proceeds to that for the more conventional case of monodirectional illumination. The azimuthal dependence of the field is separated in the usual manner by an harmonic expansion, leaving a problem in four dimensions (τ=optical depth, τ₀=thickness, ζ, η=directions of incidence and scattering) which, as is well known, is numerically extremely inconvenient. Two auxiliary radiative transfer formulations of increasing dimensionality are considered: (i) a transfer equation for the newly introduced functionb ^m(τ,η,τ₀) with Sobolev's functionΦ ^m(τ,τ₀) playing the role of a source-function. Because the incident direction does not intervene,Φ ^m is simply expressed as a single integral term involvingb ^m. For bottom illumination, an analogous equation holds for the other new functionh ^m(τ,η,τ₀). However, simple reciprocity relations link the two functions so that it is only necessary to considerb ^m; (ii) a transfer equation for the other new functiona ^m(τ,η,ζ,τ₀) with a source-function provided by Sobolev's functionD ^m(τ,ζ,τ₀). For bottom illumination, another functionf ^m(τ,η,ζ,τ₀) is introduced; by a similar argument using reciprocity relations,f ^m is reduced toa ^m rendering necessary only the consideration ofa ^m. However, a fundamental decomposition formula is obtained which shows thata ^m is expressible algebraically in terms of functions of a single angular variable. The functionsΦ ^m andD ^m are shown to be the values in the horizontal plane ofb ^m anda ^m, respectively. The other auxiliary functionsX ^m andY ^m are also expressed algebraically in terms ofb ^m. These results enable one to proceed to the final step of evaluating the radiation field for monodirectional illumination. The above reductions toalgebraic relations involving only the functionb ^m appear to be more advantageous than Sobolev's (1972) recent approach; they also circumvent some basic numerical difficulties in it. We believe the present approach may likewise prove to be superior to most (if not all) other methods of solution known heretofore.