A radiative transfer model for planetary atmospheres

The method of matched asymptotic expansions is applied to the analysis of infrared radiative energy transfer within a planetary atmosphere. The goal is to illustrate, by means of a simple mathematical model, qualitative features of atmospheric thermal structure. The matched asymptotic expansion is employed in the upper portion of the atmosphere, and the analysis illustrates that the temperature distribution in the lower portion of the atmosphere is independent of both concentration of absorbing gas and rotational line structure of the vibrational-rotation bands.     

Analysis of vector radiative transfer through a finite slab of a stochastic precipitation medium

The formulation and solution of the vector radiative transfer equation in a finite slab of a stochastic precipitation medium of binary rain rates are considered. The electromagnetic wave is supposed to encounter alternating layered segments of the two precipitation media, each with a deterministic rain rate. Both the backscattering coefficient and the bistatic coefficient are derived by taking an ensemble average of the iterative solution for the deterministic vector radiative transfer equation. Computer simulations are given to verify the solutions via the Monte Carlo method, to feature the distinctiveness of stochastic precipitation systems, and to illustrate the relationship between the stochastic parameters and the final results. It is also concluded from the computer simulations that a finite slab of a stochastic precipitation medium could be treated as an average rain-rate precipitation layer with an acceptable approximation.     

Radiative transfer in finite inhomogeneous plane-parallel atmospheres

The F_N method is used to deduce accurate numerical results for the exit distributions of radiation relevant to a finite, plane-parallel atmosphere with an exponentially varying albedo for single scattering.     

Direct solution of the radiative transfer equation for plane-parallel atmospheres

A method is described for solving the monochromatic radiative transfer equation for the case of inhomogeneous, plane-parallel scattering and absorbing atmospheres illuminated by external as well as internal sources. The solution procedure, which is based on a series expansion of the radiation intensity with respect to the angular and spatial coordinates, is analytical in nature and can thus be implemented on small computing facilites. Test calculations were performed for isotropic and Rayleigh scattering atmospheres of various optical thicknesses and single scattering albedos. The results coincide well with data from other methods given in the literature.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

A quasi-analytic solution of the radiative transfer equation for three-dimensional atmospheres

In this study, we present a new solution of the three-dimensional (3-D) radiation transfer equation (RTE). The solution employs a discretization technique to separate the independent variables involved in the 3-D RTE, and the doubling-adding method to solve the RTE explicitly and quasi-analytically. The remarkable feature of the present solution is the application of scaling-function expansion to those terms that are dependent on horizontal coordinates. Scaling-function expansion is suitable for representing irregular horizontal inhomogeneities with small-scale variations. By applying scaling-function expansion, the 3-D RTE can be formulated in the form of a vector-matrix differential equation; matrices involved in the equation are generally sparse and dominantly diagonal matrices, and this considerably reduces the labor involved in matrix calculations. We tested the performance of the present solution via radiative transfer calculations of solar radiation in horizontally inhomogeneous two-dimensional cloud models. The calculated results indicate that even if the resolution of the scaling-function expansion is too coarse in regions around small-scale variations, the influence does not spread problematically to other regions far from the variations; this illustrates the advantage of the scaling-function expansion. The present solution can be used to investigate quantitatively and to estimate the effects of cloud spatial inhomogeneity on the corresponding radiation field.     

Source function expansion method for radiative transfer in a two-layer slab

Radiative heat transfer in an isotropically scattering, absorbing and emitting two-layer slab with specularly reflecting boundaries is solved by expanding the source function by Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The reflectivity and the transmissivity of the slab for an externally incident isotropic radiation are determined. The S-1 solution yields results which are sufficiently accurate for most engineering applications.     

Two-dimensional radiative transfer in a finite scattering planar medium

The source function, radiative flux, and intensity at the boundaries are calculated for a two-dimensional, scattering, finite medium subjected to collimated radiation. The scattering phase function is composed of a spike in the forward direction super-imposed on an isotropic background. Exact radiative transfer theory is used to formulate the problem and Ambarzumian's method is used to obtain results. Using the principle of superposition, the results for any step variation in incident radiation are expressed in terms of universal functions for the semi-infinite step case. Two-dimensional effects are most pronounced at large optical thicknesses and albedos.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres

A recently developed matrix formulation of the discrete ordinate method is extended for application to an inhomogeneous atmosphere. The solution yields fluxes, as well as the complete azimuthal dependence of the intensity at any level in the atmosphere. The numerical aspects of the solution are discussed and numerical verification is provided by comparing computed results with those obtained by other methods. In particular, it is shown that a simple scaling scheme, which removes the positive exponentials in the coefficient matrix when solving for the constants of integration, provides unconditionally stable solutions for arbitrary optical thicknesses. An assessment of the accuracy to be expected is also provided, and it is shown that low-order discrete ordinate approximations yield very accurate flux values.     

The direct solution of the radiative transfer equation for optically thick atmospheres

A simple improvement of a previous direct method of solution of the spherical harmonics approximation to the radiative transfer equation results in higher computational efficiencies by factors of 2–4; these higher efficiencies are particularly important to shorten the lengthy computations required in optically thick nonhomogeneous atmospheres.     

A simple computational method for internal polarized radiation fields of finite slab atmospheres

An extended doubling method is formulated, which provides together with the emergent radiation also the internal polarized radiation field without additional iterations. Two sets of linear regular integral relations are derived, which have to be fulfilled by the surface Green's function matrix or, equivalently, by the Stokes vector of the slab albedo problem radiation field. The integral relations refer to the half range angular variable of the direction of incidence and to the full range angular variable of the direction of light propagation, respectively.     

Time-dependent radiation transfer with Rayleigh scattering in finite slab media

The time-dependent radiation transfer equation in a finite plane geometry with Rayleigh scattering is studied. The traveling wave transformation is used to obtain the corresponding stationary-like equation. Pomraning-Eddington approximation is then used to find the solution. Numerical results for reflectivity at the left boundary and transmissivity from the right boundary are presented at different times. The medium is assumed to have specular-reflecting boundaries with angular-dependent externally incident flux. Two different weight functions are introduced to force the boundary conditions to fulfill.     

A direct solution of the radiative transfer equation: Application to Rayleigh and Mie atmospheres

A direct method is given for the solution of the spherical harmonics approximation to the equation of radiative transfer in plane-parallel atmospheres. Although the method is formulated theoretically for non-homogeneous atmospheres with an arbitrary phase function, at present it has only been implemented for homogeneous atmospheres. Test computations performed for Rayleigh and Mie scattering phase functions show that the direct method is unconditionally stable and solves efficiently problems both for optically thin and very thick atmospheres. Timing comparisons with the method of Chandrasekhar for Rayleigh atmospheres and with an integral-equation iterative method for Mie atmospheres are quite favorable to the proposed method.     

Hybrid finite volume/ finite element method for radiative heat transfer in graded index media

The rays propagate along curved path determined by the Fermat principle in the graded index medium. The radiative transfer equation in graded index medium (GRTE) contains two specific redistribution terms (with partial derivatives to the angular coordinates) accounting for the effect of the curved ray path. In this paper, the hybrid finite volume with finite element method (hybrid FVM/FEM) (P.J. Coelho, J. Quant. Spectrosc. Radiat. Transf., vol. 93, pp. 89–101, 2005) is extended to solve the radiative heat transfer in two-dimensional absorbing-emitting-scattering graded index media, in which the spatial discretization is carried out using a FVM, while the angular discretization is by a FEM. The FEM angular discretization is demonstrated to be preferable in dealing with the redistribution terms in the GRTE. Two stiff matrix assembly schemes of the angular FEM discretization, namely, the traditional assembly approach and a new spherical assembly approach (assembly on the unit sphere of the solid angular space), are discussed. The spherical assembly scheme is demonstrated to give better results than the traditional assembly approach. The predicted heat flux distributions and temperature distributions in radiative equilibrium are determined by the proposed method and compared with the results available in other references. The proposed hybrid FVM/FEM method can predict the radiative heat transfer in absorbing-emitting-scattering graded index medium with good accuracy.     

Stochastic radiative transfer in a finite plane-parallel medium with general boundary conditions

The time-independent radiative transfer problem in a scattering and absorbing planar random medium with general boundary conditions and internal energy source is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The problem is solved in terms of the solution of the corresponding free-source problem with simple boundary conditions which is solved using Pomraning-Eddington approximation in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average partial heat fluxes are calculated in terms of the albedoes of the source-free problem. Results are obtained for isotropic and anisotropic scattering for specular and diffused reflecting boundaries.     

Numerical results for finite order X and Y functions of radiative transfer

It is well known that, in the theory of radiative transfer, Chandrasekhar's X and Y functions play an important role in the diffuse reflection and transmission problem (cf. Chandrashekhar⁽¹⁾). In a preceding paper (cf. Bellmanet al.⁽¹⁰⁾), graphs and selected tables of these functions covering wide ranges of slab thickness and albedos for single scattering have been provided. In this paper, making use of a system of coupled integral recurrence relations for finite order X and Y functions (cf. Bellmanet al.⁽¹⁴⁾), numerical results for these basic functions are tabulated up to optical thickness τ = 2.0 from τ = 0.1, assuming the conservative case of isotropic scattering. The maximum order of these functions is taken to be fifteenth. It is shown that the accuracy obtained is satisfactory in the domain under consideration. Furthermore, numerical results for Chandrasekhar's approximation for X and Y functions are also tabulated for stabs of small optical thickness.     

A fourier transform solution for radiative transfer in a slab with isotropic scattering and boundary reflection

The Fourier transform method is used to formulate an exact solution for radiative heat transfer in an isotropically scattering, absorbing and emitting plane-parallel slab with specularly reflecting boundaries. The analysis leads to a solution involving expansion coefficients which can be determined from a matrix equation to the desired degree of accuracy. The numerical results obtained for the hemispherical reflectivity and transmissivity show that even the lowest order solution gives good accuracy.     

Three-dimensional radiative transfer modeling using the control volume finite element method

In the present study, a three-dimensional algorithm for the treatment of radiative heat transfer in emitting, absorbing and scattering media is developed. The approach is based on the utilization of control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied at the first time to 3D radiative heat transfer in participating media. The accuracy of the present algorithm is tested by comparing its predictions to other published works. Comparisons show that CVFEM produces good results. Moreover, this approach permits compatibility with other numerical methods used for computational fluids mechanics problems.     

An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres

The three-dimensional (3D) diffusion radiative transfer equation, which utilizes a four-term spherical harmonics expansion for the scattering phase function and intensity, has been efficiently solved by using the full multigrid numerical method. This approach can simulate the transfer of solar and thermal infrared radiation in inhomogeneous cloudy conditions with different boundary conditions and sharp boundary discontinuity. The correlated k-distribution method is used in this model for incorporation of the gaseous absorption in multiple-scattering atmospheres for the calculation of broadband fluxes and heating rates in the solar and infrared spectra. Comparison of the results computed from this approach with those computed from plane-parallel and 3D Monte Carlo models shows excellent agreement. This 3D radiative transfer approach is well suited for radiation parameterization involving 3D and inhomogeneous clouds in climate models.