Allen C. Cogley

Exponential Approximation for Daily Average Solar Heating or Photolysis

Abstract Formulations of instantaneous solar heating or photolytic rates, for various atmospheric absorbers, as functions of altitude and sun angle are available in the literature. Such work can be integrated over the solar day to obtain the net daily effect. When incorporating these processes in long-range atmospheric forecasting models with large time steps or when assuming steady state, it may be desirable to replace the time integrals by daily average rates that are simple functions of latitude and season (sun declination). To accomplish this the integral over the solar day, which is shown to have the form of a modified exponential-integral function, is approximated by a pure exponential. This gives a daily average rate as a multiplication factor times the instantaneous rate evaluated at an appropriate sun angle. The multiplication factor and sun angle are analytically found by matching certain properties of the exponential-integral and exponential functions. The result is several choices for the fitt...

A fundamental-source-function formulation of radiative transfer and the resulting fundamental reciprocity relations

Abstract An important problem in radiative transfer is finding the radiative fields produced by various illuminations (both external and internal) of a plane-parallel, inhomogeneous, absorbing, emitting, and anisotropically-scattering finite medium. One approach to a solution is to find the source function, which represents the rate of production of scattered radiation per unit volume and solid angle, at each point in the medium. The present study develops the existence of a Greens function, called the fundamental source function, which separates the optical properties of the medium from the driving illumination. Radiative linearity then allows the representation of all possible source functions as convolutions of the illumination with the fundamental source function. Parametric differentiation (invariant imbedding) is used to replace the governing linear integral equation for the fundamental source function with a set of differential equations appropriate for numerical integration. This approach for finding the fundamental source function leads naturally to the introduction of fundamental scattering and transmission functions. Our inclusion of anisotropic internal illumination (sources) allows us to develop four new reciprocity relations involving these functions. The reciprocity relations state general equivalences between an internally and an externally driven medium and thus greatly reduce the complexity of radiative transfer.

Adding and invariant imbedding equations in matrix notation for all the scattering functions

Abstract Previous work by the author introduced a radiative formulation, containing a delta interior illumination, that allowed scattering solutions driven by internal sources to be handled in complete analogy to those for the standard problem (external delta illumination scattering through a medium). This analogy was made explicit by defining the three levels of scattering functions, S s -level, S s - and S F -level, and S F -level, that characterize scattering through, into and out of, and within a finite medium, respectively. For an inhomogeneous medium the invariant imbedding method was employed to solve for these functions. This paper continues the work by showing that: (1) Adding equations can be derived for all the scattering functions using one superposition formula. (2) Adding and invariant imbedding computational methods are closely related and should be used in combination for efficient calculations. (3) A new set of functions can be defined that represent scattering out of a medium driven by thermal sources. (4) All scattering functions can be converted to represent a planetary problem by one adding step. References are given for numerical results using this formulation.

Derivation and application of the reciprocity relations for radiative transfer with internal illumination

Abstract A Greens function formulation is used to derive basic reciprocity relations for planar radiative transfer in a general medium with internal illumination. Reciprocity (or functional symmetry) allows an explicit and generalized development of the equivalence between source and probability functions. Assuming similar symmetry in three-dimensional space, a general relationship is derived between planar-source intensity and point-source total directional energy. These quantities are expressed in terms of standard (universal) functions associated with the planar medium, while all results are derived from the differential equation of radiative transfer.

Scattering of emitted radiation from inhomogeneous and nonisothermal layers

Abstract A parametric study is performed for the exiting monochromatic intensities scattered from finite, plane-parallel, inhomogeneous layers that are driven solely by a distribution of thermal sources. Intensities are obtained by invariantly imbedding the standard and thermal scattering functions. The single scattering albedo ω and the Henyey-Greenstein phase-function parameter g are varied independently, and both linear and exponential profiles are considered. Linear temperature profiles are used, including temperature inversions. The resulting intensities I(μ), μ representing the direction cosine of propagation, are discussed from a remote sensing point of view. For an isothermal and homogeneous medium, the gross characteristics of I(μ) represented by its overall slope I(0)/I(1), mean value (magnitude), and an interior maximum value can be related to the total optical depth t0, ω, and g, respectively. For a homogeneous medium, linearly decreasing (in the line of sight) temperature profiles tend to obscure the g information and decrease the apparent optical depth. On the other hand, linearly increasing temperature profiles tend to retain g information and increase the apparent optical depth. Temperature inversion profiles give intensities very similar to those for purely linear profiles. Linear and exponential variations of both ω and g for constant temperatures give similar intensity fields. Results for a variation in g can be reproduced fairly well with an average g value. This cannot be done, however, for ω profiles.

Numerical results for the thermal scattering functions

A recent formulation in radiative transfer defined the thermal scattering functions that characterize radiative transfer from a general, plane-parallel, finite medium driven solely by an internal distribution of thermal sources. Exiting diffuse intensities are expressed as space convolutions of the thermal scattering functions with any thermal source distribution. A parametric study is presented to obtain the basic structure of these scattering functions. The independent variables of these azimuthally independent functions are the direction consine μ and source location t, while the parameters are the single scattering albedo ω, total optical depth t0, and the asymmetry factor g in the Henyey-Greenstein phase function. The basic functional trends are discussed using various parametric plots, and selected tabular results are given to allow numerical checks. The computational method is invariant imbedding. As a particular application, these functions are used in the following companion paper to obtain exiting intensities from inhomogeneous and nonisothermal media.

Radiative transport of Lorentz lines in nonisothermal gases

Abstract A modified Ladenburg-Reiche analysis is developed for predicting nonisothermal curves of growth for isolated, unshifted Lorentz lines. The superposition of Lorentz lines along nonisothermal paths is approximated by an equivalent Lorentz line, whose two fitting parameters are internally defined. The formulation is exact in the weak- and strong-line limits and is a valid approximation for all optical depths. Its accuracy is investigated for a family of linear variations in the line strength and half-half-width, for which exact results can be obtained analytically. The error in the nonisothermal equivalent width is strongly dependent on variations in the half-half-width and only weakly dependent on variations in the line strength. For a ten-fold change in the half-half-width over the optical path, a maximum error of only three per cent is observed for the equivalent width.

Radiative heat transfer in a completely general plane-parallel environment

Abstract A new approach to radiative heat transfer with scattering is presented and used to obtain the first general solution for radiative equilibrium in a non-grey, plane-parallel medium. Use of the method when including conduction and convection is also discussed. Diffuse radiative fields are calculated in terms of defined scattering functions (Greens functions) that represent the response of the medium and any scattering (reflecting) surfaces to unitary-type illumination. These scattering functions are found using the computationally fast adding/doubling method. The energy conservation equation containing these scattering functions is then solved numerically for any particular heat transfer problem.

Error analysis of the adding/doubling method for radiative scattering in inhomogeneous media

Abstract The doubling method is a fast and exact procedure for calculating radiative transfer in a homogeneous, scattering, plane-parallel medium. It can also be used in the adding mode for an inhomogeneous medium that is approximated by a finite number of homogeneous sublayers with different radiative properties. The errors caused by this approximation are analyzed in this paper through comparison with invariant imbedding calculations that are slow but exact for inhomogeneous media. A procedure is developed so errors can be estimated and controlled when using the faster adding/doubling calculations.

Exploiting the linearity of radiative transfer

Abstract A standard problem in radiative transfer is finding the external and internal radiative fields produced by uniform, parallel rays illuminating the top of a one-dimensional, scattering and absorbing medium of finite optical thickness. This problem has been solved in several ways with various physical restrictions. One approach is by finding the source function that represents the rate of production of scattered radiation per unit volume per unit solid angle at each point in the medium. The present paper develops and uses the idea that the standard source function is an influence function for a given medium. The linearity of radiative transfer is then used to find certain general source functions in terms of the standard one. The usefulness of the above concept is demonstrated by the following four problems: (1) derivation of Chandrasekhars four principles of invariance from the radiative transfer equation, (2) derivation of the equations governing Chandrasekhars X - and Y - functions without using the invariance principles or resolvent kernels, (3) finding the source function for a medium with a Lamberts-law bottom, and (4) finding the source function for a medium with a bottom that is a perfect specular reflector.