Nonisothermal band model theory

Abstract

Theoretical formulations of radiation band models for general nonuniform optical paths are presented. The models are framed within the statistical band model for an array of Lorentz lines and an inverse line strength distribution. Radiative transfer for an isolated line in a general nonuniform medium and the statistical model for a uniform optical path are reviewed in order to provide the foundation required for the band model formulations. Two approaches to the development of these models are taken. The first is based on making approximations to such radiative transfer functions as transmittance or equivalent width and yields models equivalent or similar to the traditional Curtis-Godson approximation. The second treats approximations to the spatial derivatives of these functions. From the standpoint of computing line or band radiance, the spatial derivatives are more fundamental quantities than the transfer functions themselves; consequently, these latter “derivative approximations” are instrinsically more accurate than the Curtis-Godson type approximations. All of the models are formulated to give the spatial derivative of the mean equivalent width function W ̄ (s) δ in the form 1 δ d W ̄ (s) ds =c(s)p(s) k ̄ (s)y(s) , where c( s), p( s) and k̄( s) are the concentration, total pressure and absorption band model parameter, respectively, at the local path position s. The functional form of the derivative function y( s) is derived for each of the models. In general, y( s) is a function of both local and path-averaged (subscript e) band model parameters. These local and averaged parameters are used to define a dimensionless optical depth parameter x e = u k ̄ e β e and three nonuniformity indexes ρ= γ ̄ γ ̄ e , r= β β e and θ = δ ̄ e δ ̄ (γ̄ is line width, ḡd is the mean line spacing band model parameter, and β = 2μ γ ̄ γ ̄ ). The explicit manner in which these parameters enter into the various functional forms for y( s) is derived and discussed. The introduction of the parameter q is an important aspect of the new models and is an explicit measure of nonisothermality along the optical path. In addition to the traditional Curtis-Godson definitions for the effective parameters k̄ e and β e , new definitions for the path averages γ̄ e and δ̄ e are derived. These new definitions are obtained from the fundamental properties of the assumed inverse line strength distribution. A summary of all the models is given as a table of relevant equations for use in practical calculations.