An Iterative Solution for Anisotropic Radiative Transfer in a Slab

1979 ◽  
Vol 101 (4) ◽  
pp. 695-698 ◽  
Author(s):  
W. H. Sutton ◽  
M. N. O¨zis¸ik
Keyword(s):  
Radiative Transfer ◽  
Iterative Method ◽  
Integral Form ◽  
Iterative Solution ◽  
Single Scattering ◽  
Anisotropic Scattering ◽  
Initial Guess ◽  
Isotropic Scattering ◽  
Wide Range ◽  
Reflecting Boundaries

An iterative method is applied to solve the integral form of the equation of radiative transfer for the cases of isotropic scattering, highly forward, and backward anisotropic scattering in plane-parallel slab with reflecting boundaries. Calculations are performed for the values of single scattering albedo from ω = 0.7 to 1.0 where the convergence was previously reported to be poor. It is found that the convergence is significantly improved for most cases if the P-1 approximation of the spherical harmonics method is used for the initial guess. Results are presented for the hemispherical reflectivity and transmissivity of the slab over a wide range of parameters.

scholarly journals Using a Parameterization of a Radiative Transfer Model to Build High-Resolution Maps of Typical Clear-Sky UV Index in Catalonia, Spain

2005 ◽  
Vol 44 (6) ◽  
pp. 789-803 ◽  
Author(s):  
Jordi Badosa ◽  
Josep-Abel González ◽  
Josep Calbó ◽  
Michiel van Weele ◽  
Richard L. McKenzie
Keyword(s):  
High Resolution ◽  
Radiative Transfer ◽  
Zenith Angle ◽  
Solar Zenith Angle ◽  
Absolute Error ◽  
Single Scattering ◽  
Radiative Transfer Model ◽  
Transfer Model ◽  
Uv Index ◽  
Wide Range

Abstract To perform a climatic analysis of the annual UV index (UVI) variations in Catalonia, Spain (northeast of the Iberian Peninsula), a new simple parameterization scheme is presented based on a multilayer radiative transfer model. The parameterization performs fast UVI calculations for a wide range of cloudless and snow-free situations and can be applied anywhere. The following parameters are considered: solar zenith angle, total ozone column, altitude, aerosol optical depth, and single-scattering albedo. A sensitivity analysis is presented to justify this choice with special attention to aerosol information. Comparisons with the base model show good agreement, most of all for the most common cases, giving an absolute error within ±0.2 in the UVI for a wide range of cases considered. Two tests are done to show the performance of the parameterization against UVI measurements. One uses data from a high-quality spectroradiometer from Lauder, New Zealand [45.04°S, 169.684°E, 370 m above mean sea level (MSL)], where there is a low presence of aerosols. The other uses data from a Robertson–Berger-type meter from Girona, Spain (41.97°N, 2.82°E, 100 m MSL), where there is more aerosol load and where it has been possible to study the effect of aerosol information on the model versus measurement comparison. The parameterization is applied to a climatic analysis of the annual UVI variation in Catalonia, showing the contributions of solar zenith angle, ozone, and aerosols. High-resolution seasonal maps of typical UV index values in Catalonia are presented.

An Approximate Solution to Radiative Transfer in Two-Dimensional Rectangular Enclosures

1997 ◽  
Vol 119 (4) ◽  
pp. 738-745 ◽  
Author(s):  
J. B. Pessoa-Filho ◽  
S. T. Thynell
Keyword(s):  
Radiative Transfer ◽  
Iterative Solution ◽  
Anisotropic Scattering ◽  
Solution Procedure ◽  
Two Dimensional ◽  
Scattering Media ◽  
Discontinuous Nature ◽  
Selection Of ◽  
Iterative Solution Procedure ◽  
Numerical Approaches

The application of a new approximate technique for treating radiative transfer in absorbing, emitting, anisotropically scattering media in two-dimensional rectangular enclosures is presented. In its development the discontinuous nature of the radiation intensity, stability of the iterative solution procedure, and selection of quadrature points have been addressed. As a result, false scattering is eliminated. The spatial discretization can be formed without considering the chosen discrete directions, permitting a complete compatibility with the discretization of the conservation equations of mass, momentum, and energy. The effects of anisotropic scattering, wall emission, and gray-diffuse surfaces are considered for comparison with results available in the literature. The computed numerical results are in excellent agreement with those obtained by other numerical approaches.

scholarly journals A Theoretical Model of Radiative Transfer in Young Sea Ice

1982 ◽  
Vol 28 (99) ◽  
pp. 341-356
Author(s):  
Donald K. Perovich ◽  
Thomas C. Grenfell
Keyword(s):  
Theoretical Model ◽  
Radiative Transfer ◽  
Phase Function ◽  
Incident Radiation ◽  
Single Scattering ◽  
Anisotropic Scattering ◽  
Discrete Ordinates ◽  
Ice Thickness ◽  
Surface Layers ◽  
Beam Incident

AbstractA four stream discrete-ordinates photometric model including both anisotropic scattering and refraction at the boundaries is presented which treats the case of a floating ice slab. The effects of refraction and reflection on the redistribution of the incident radiation field as it enters the ice are examined in detail. Using one- and two-layer models, theoretical albedos and transmittances are compared to values measured in the laboratory for thin salt ice. With an experimentally determined three-parameter Henyey–Greenstein phase function, comparisons at 650 nm yield single-scattering albedos ranging from 0.95 to 0.9997. The models are then used to compare the effects of diffuse and direct-beam incident radiation, to investigate the dependence of spectral albedo and transmittance on ice thickness, and to determine the influence of very cold and melted surface layers.

scholarly journals A Theoretical Model of Radiative Transfer in Young Sea Ice

1982 ◽  
Vol 28 (99) ◽  
pp. 341-356 ◽  
Author(s):  
Donald K. Perovich ◽  
Thomas C. Grenfell
Keyword(s):  
Theoretical Model ◽  
Radiative Transfer ◽  
Phase Function ◽  
Incident Radiation ◽  
Single Scattering ◽  
Anisotropic Scattering ◽  
Discrete Ordinates ◽  
Ice Thickness ◽  
Surface Layers ◽  
Beam Incident

AbstractA four stream discrete-ordinates photometric model including both anisotropic scattering and refraction at the boundaries is presented which treats the case of a floating ice slab. The effects of refraction and reflection on the redistribution of the incident radiation field as it enters the ice are examined in detail. Using one- and two-layer models, theoretical albedos and transmittances are compared to values measured in the laboratory for thin salt ice. With an experimentally determined three-parameter Henyey–Greenstein phase function, comparisons at 650 nm yield single-scattering albedos ranging from 0.95 to 0.9997. The models are then used to compare the effects of diffuse and direct-beam incident radiation, to investigate the dependence of spectral albedo and transmittance on ice thickness, and to determine the influence of very cold and melted surface layers.

A Source Function Expansion in Radiative Transfer

1980 ◽  
Vol 102 (4) ◽  
pp. 715-718 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
W. H. Sutton
Keyword(s):  
Radiative Transfer ◽  
Source Function ◽  
Transfer Problem ◽  
Integral Form ◽  
Computer Time ◽  
Algebraic Equations ◽  
Function Expansion ◽  
Expansion Technique ◽  
Reflecting Boundaries ◽  
Special Case

The radiative heat transfer problem for an isotropically scattering slab with specularly reflecting boundaries is reduced to the solution of a set of algebraic equations by expanding the source function in Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The lowest order S-1 analysis requires very little computer time for calculations, is easy to apply and yields results which are sufficiently accurate. For an absorbing, emitting, isotropically scattering medium with small and intermediate optical thickness (i.e., τ = 2), which is of great interest in engineering applications, and for which the P-1 and P-3 solutions of the P-N method are not sufficiently accurate, the S-1 solution yields highly accurate results. In the case of a slab with diffusely reflecting boundaries, the problem is split up into a set of simpler problems each of which is solved with the source function expansion technique as a special case of the general problem considered.

Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Jing Ma ◽  
Ya-Song Sun ◽  
Ben-Wen Li
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Discrete Ordinates ◽  
Spectral Collocation ◽  
Graded Index ◽  
Wide Range ◽  
Derivative Term ◽  
Scattering Phase Function

A completely spectral collocation method (CSCM) is developed to solve radiative transfer equation in anisotropic scattering medium with graded index. Different from the Chebyshev collocation spectral method based on the discrete ordinates method (SP-DOM), the CSCM is used to discretize both the angular domain and the spatial domain of radiative transfer equation. In this approach, the angular derivative term and the integral term are approximated by the high order spectral collocation scheme instead of the low order finite difference approximations. Compared with those available data in literature, the CSCM has a good accuracy for a wide range of the extinction coefficient, the scattering albedo, the scattering phase function, the gradient of refractive index and the boundary emissivity. The CSCM can provide exponential convergence for the present problem. Meanwhile, the CSCM is much more economical than the SP-DOM. Moreover, for nonlinear anisotropic scattering and graded index medium with space-dependent albedo, the CSCM can provide smoother results and mitigate the ray effect.

Resolved Order of Scattering for the Solution of Radiative Transfer Equation

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Liangyu Wang ◽  
Robert J. Hall ◽  
Meredith B. Colket
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Scattering Media ◽  
Radiative Intensity ◽  
Reflecting Boundaries ◽  
Strong Scattering

The solution of the radiative transfer equation (RTE) becomes complicated when the participating medium is scattering and/or the boundary walls are reflecting. To reduce the complexity, the resolved order of scattering (ROS) formulation described in this paper separates the radiative intensities being solved by RTE into a series of intensities corresponding to different orders of the scattering and reflection events. The resulting equation of transfer for each order of radiative intensity is not only much simpler to solve but also represents the physical scattering/reflection processes that are hidden in the original full RTE. The ROS formulation provides a mathematically rigorous and elegant means of solving RTE for strong scattering media with or without reflecting boundaries. Sample calculations are presented for a droplet-laden, 3D enclosure with strong anisotropic scattering.

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