Stochastic radiative transfer in multilayer broken clouds. Part I: Markovian approach

A two-part paper describes the statistical treatment of solar radiative transfer in multilayer broken clouds. The proposed approach is a logical development of the statistical ones originally suggested for a single-layer broken clouds. This first part introduces a new statistically inhomogeneous Markovian model that allows one to properly account for different combinations of the random and maximum overlap of broken clouds at distinct vertical layers. The statistically inhomogeneous Markovian model and the stochastic radiative transfer equation have been used to derive equations for the mean radiance of solar radiation. It was demonstrated that in extreme cases the obtained equations agree with corresponding equations previously derived for (i) the statistically homogeneous broken clouds and (ii) the vertically inhomogeneous overcast clouds.     

A Monte Carlo method for 3D thermal infrared radiative transfer

A 3D Monte Carlo model for specific application to the broadband thermal radiative transfer has been developed in which the emissivities for gases and cloud particles are parameterized by using a single cubic element as the building block in 3D space. For spectral integration in the thermal infrared, the correlated k-distribution method has been used for the sorting of gaseous absorption lines in multiple-scattering atmospheres involving 3D clouds. To check the Monte-Carlo simulation, we compare a variety of 1D broadband atmospheric fluxes and heating rates to those computed from the conventional plane-parallel (PP) model and demonstrate excellent agreement between the two. Comparisons of the Monte Carlo results for broadband thermal cooling rates in 3D clouds to those computed from the delta-diffusion approximation for 3D radiative transfer and the independent pixel-by-pixel approximation are subsequently carried out to understand the relative merits of these approaches.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.     

The opacity coefficient method—nongray determination of inhomogeneous atmospheric extinction

The computation of radiation transmittance in nongray, inhomogeneous atmospheric models is frequently complicated by complex bands of line spectra which range in value over many orders of magnitudes and depend strongly on either or both of pressure and temperature. We present here a new opacity sampling technique which is shown to determine correctly the wavelength-averaged extinction due to path-dependent realizations of banded line spectra. The technique is easy to implement computationally and is applicable to a wide variety of atmospheric problems in which frequent iteration of the radiative transfer model is required. We consider two such instances: modeling of solar flux attenuation for use in a time-dependent planetary ionosphere model and retrieval from nadir measurements of backscattered solar irradiance. The power of the new method lies in its straightforward analytical treatment of both atmospheric inhomogeneity and spectral complexity. It is thus relevant for both retrieval and radiative transfer modeling purposes.     

A numerical model for multigroup radiation hydrodynamics

We present in this paper a multigroup model for radiation hydrodynamics to account for variations of the gas opacity as a function of frequency. The entropy closure model (M₁) is applied to multigroup radiation transfer in a radiation hydrodynamics code. In difference from the previous grey model, we are able to reproduce the crucial effects of frequency-variable gas opacities, a situation omnipresent in physics and astrophysics. We also account for the energy exchange between neighbouring groups which is important in flows with strong velocity divergence. These terms were computed using a finite volume method in the frequency domain. The radiative transfer aspect of the method was first tested separately for global consistency (reversion to grey model) and against a well-established kinetic model through Marshak wave tests with frequency-dependent opacities. Very good agreement between the multigroup M₁ and kinetic models was observed in all tests. The successful coupling of the multigroup radiative transfer to the hydrodynamics was then confirmed through a second series of tests. Finally, the model was linked to a database of opacities for a Xe gas in order to simulate realistic multigroup radiative shocks in Xe. The differences with the previous grey models are discussed.     

Discrete vs. continuum-scale simulation of radiative transfer in semitransparent two-phase media

The mathematical formulation of the continuum approach to radiative transfer modeling in two-phase semi-transparent media is numerically validated by comparing radiative fluxes computed by (i) direct, discrete-scale and (ii) continuum-scale approaches. The analysis is based on geometrical optics. The discrete-scale approach uses the Monte Carlo ray-tracing applied directly to real 3D geometry measured by computed tomography. The continuum-scale approach is based on a set of continuum-scale radiative transfer equations and associated radiative properties, and employs the Monte Carlo ray-tracing for computations of radiative fluxes and for computations of the radiative properties. The model two-phase media are reticulate porous ceramics and a particle packed bed, each composed of semitransparent solid and fluid phases. The results obtained by the two approaches are in good agreement within the limits of statistical uncertainty. The continuum-scale approach leads to a reduction in computational time by approximately one order of magnitude, and is therefore suited to treat radiative transfer problems in two-phase media in a wide range of engineering applications.     

A diffusion-based approximate model for radiation heat transfer in a solar thermochemical reactor

An approximate numerical method for fast calculations of the radiation heat transfer in a solar thermochemical reactor cavity is formulated based on the separate treatment of the solar and thermal radiative exchange by the diffusion approach. The usual P₁ approximation is generalized by applying an equivalent radiation diffusion coefficient for the optically thin central part of the cavity. The resulting boundary-value problems are solved using the finite element algorithm. The accuracy of the model is assessed by comparing the results to those obtained by a pathlength-based Monte Carlo simulation. The applicability of the proposed model is demonstrated by performing calculations for an example problem, which incorporates a range of parameters typical for a solar chemical reactor and the spectral radiative properties of polydisperse zinc oxide particles.     

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.     

Layer-edge conditions for multislab atmospheric radiative transfer problems

We derive nonstandard layer-edge conditions for efficient solution of multislab atmospheric radiative transfer problems. We begin by defining a local radiative transfer problem on the lowermost layer of a multislab model atmosphere and we consider a standard discrete ordinates version of this local problem. We then make use of a recently developed computational method in order to derive layer-edge conditions involving incident, reflected and transmitted radiation. These layer-edge conditions for the lowermost layer are given in terms of inherent optical properties of the layer, the solar zenith angle and the quadrature set used in the discrete ordinates approach. They can be used to increase the efficiency of our computational method in solving practical problems in atmospheric radiative transfer. Moreover, they are amenable to incorporation into other discrete ordinates methods. To illustrate, we report numerical results for two atmospheric model problems.     

Influence of surface roughness on the reflective properties of snow

In this paper the influence of 3D effect on snow reflection function (SRF) and albedo is studied in the framework of the stochastic radiative transfer theory. In particular, the corresponding equations for the averaged intensity of reflected light are solved for the ensemble of realizations of the stochastic field κ(r), describing the distribution of 3D elements on the flat semi-infinite snow layer (SISL). The reflection from the underlying SISL is modeled using the solution of the 1D radiative transfer equation. The corresponding look-up tables were compiled beforehand and used in the simulation process. In accordance with the previous studies, it was found that the albedo of snow layer is reduced (in particular, in the infrared region), if 3D effects are taken into account. There is no such a reduction, if light absorption in snow is absent. The 3D effects may increase or decrease SRF depending on the sastrugi fraction and illumination/observation conditions.     

Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P₁ approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P₁ approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P₁ approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P₁ approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S₆ and S₈), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.     

Radiative transfer within the atmospheres of the major planets

A modified Eddington approximation is presented which describes radiative transfer due to the pressure-induced far infrared spectrum of hydrogen. When applied to the atmospheres of the major planets, the radiative transfer model produces atmospheric temperature profiles which are in good agreement with more detailed calculations. Solar absorption and infrared transmission by the fundamental bands of methane are further included for Jupiter and Saturn, resulting in a temperature inversion within both atmospheres above an altitude corresponding to a pressure of roughly 0.01 atm. This relatively high inversion region appears to be qualitatively consistent with infrared observations of methane and ammonia bands for Jupiter. Additional sources of solar absorption, consisting of the pressure-induced vibration-rotation bands of hydrogen and the overtones of the v₃ methane fundamental, are shown to have a minor effect upon the atmospheric thermal structure for Jupiter and Saturn.     

Cloud 3D effects on broadband heating rate profiles: I. Model simulation

Solar broadband heating directly drives the atmospheric and ocean circulations, and is largely determined by cloud spatial 3-diminesional (3D) structures. To study the cloud 3D effects on radiation, a 3D broadband Monte-Carlo radiative transfer model, along with an Independent Pixel/Column Approximation (IPA) method, is used to simulate radiation and heating rate of three typical cloud fields generated by cloud resolving models (CRM). A quantitative and statistical estimation of cloud 3D effects has been developed to investigate the impact of cloud 3D structures on both heating rate strength, STD_Bias, and vertical distribution, CorrCoef. The cloud 3D structures affect some clouds more in heating rate strength and others more in vertical distribution. It is crucial to use the combination of CorrCoef and STD_Bias for better quantitative evaluation of the 3D effects. Furthermore, there is no simple way to define a critical resolution (or average radius), within which the IPA heating rate profiles closely represent the true 3D heating rate profiles. The critical radius (or resolution) strongly depends on solar incident angle as well as cloud vertical distribution. Also, the critical radii for clear-sky columns are larger than for cloudy columns, although the corresponding STD_Bias for clear-sky columns are smaller than for cloudy columns. Analysis based on two different statistical average methods illustrates that the cloud 3D effects due to the dimensionality difference between the 3D clouds (circle average) and 2D clouds (line average) significantly impact on the heating rate profiles.     

A fast linearized pseudo-spherical two orders of scattering model to account for polarization in vertically inhomogeneous scattering-absorbing media

We calculate the reflection matrix for the first two orders of scattering in a vertically inhomogeneous, scattering-absorbing medium. We take full account of polarization and perform a complete linearization (analytic differentiation) of the reflection matrix with respect to both the inherent optical properties of the medium and the surface reflection condition. Further, we compute a scalar-vector correction to the total intensity due to the effect of polarization; this correction is also fully linearized. The solar beam attenuation has been computed for a pseudo-spherical atmosphere.Results from the two orders of scattering (2OS) model have been tested against scalar intensities for an inhomogeneous atmosphere, and against Stokes vector results for a homogeneous atmosphere. We have also performed backscatter simulations of reflected sunlight in the O₂A band for a variety of geometries, and compared our results with those from a full vector multiple scattering code. Our results are exact in the center of strong lines and most inaccurate in the continuum, where polarization is least significant. The s- and p-polarized radiances are always computed very accurately. The effect of gas absorption optical depth, solar zenith angle, viewing geometry, surface albedo and wind speed (in the case of ocean glint) on the intensity, polarization and corresponding weighting functions have been investigated. It is shown that the 2OS model provides fast and reliably accurate polarization corrections to the scalar-model radiance and weighting function fields. The model can be implemented in operational retrieval algorithms as an adjunct radiative transfer code to deal with polarization effects.     

Light reflection from a rough liquid surface including wind-wave effects in a scattering atmosphere

Visible and near-IR images of the ocean surface, taken from remote satellites, often contain important information of near-surface or sub-surface processes, which occur on, or over the ocean. Remote measurements of near surface winds, sea surface temperature and salinity, ocean color and underwater bathymetry, all, one way or another, depend on how well we understand sea surface roughness. However, in order to extract useful information from our remote measurements, we need to construct accurate models of the transfer of solar radiation inside the atmosphere as well as, its reflection from the sea surface. To approach this problem, we numerically solve the radiative transfer equation (RTE) by implementing a model for the atmosphere-ocean system. A one-dimensional atmospheric radiation model is solved via the widely known doubling and adding method and the ocean body is treated as a boundary condition to the problem. The ocean surface is modeled as a rough liquid surface which includes wind interaction and wave states, such as wave age. The model can have possible applications to the retrieval of wind and wave states, such as wave age, near a Sun glint region.     

太阳能吸热器玻璃窗辐射与导热计算

在太阳能塔式热发电空气吸热器工作过程中,吸热器玻璃窗受到聚焦太阳光、环境以及吸热面（体）的复合辐射。本文采用改进蒙特卡罗法、有限容积法,研究了太阳能空气吸热器石英玻璃窗半透明介质内部辐射与导热的耦合换热,研究了不同厚度半透明介质石英玻璃窗在工作面处于不同第三类边界条件下的稳态温度分布。研究表明石英玻璃的厚度和内侧空气温...     

A full spectrum k-distribution based non-gray radiative property model for unburnt char

By using the concept of weighted sum of four gray particles and spectrum k-distribution (WSGP-SK), a non-gray radiative property model for unburnt char particles is developed. Based on the carbon burnout kinetic model for structure during oxidation, and the linear mixed approximation theory for complex index of refraction, spectral radiative properties of unburnt char particles are first calculated as function of the burnout ratio by Mie theory. Referring to the full spectrum k-distribution model, k-distribution is applied to reorder absorption and scattering efficiencies of particles. Then, weighting factors and efficiency factors of the non-gray radiative property model are directly obtained from Gaussian integral points of k-distribution. The model is validated against the benchmark solutions of line-by-line (LBL) model. Maximum relative errors of this model are 3% and 15% for radiative heat fluxes and source terms in non-isothermal inhomogeneous particulate media, respectively. The assumption of linearly varying radiative properties with burnout ratio (Lockwood et al. 1986) will result in a predicted deviation of 53% for radiative source terms. Results also show that this non-gray model is remarkably better than the Planck mean method. Moreover, a satisfactory comparison with LBL solutions is achieved in the gas and particle mixture by combining the non-gray WSGG-SK model (Guo et al. 2015). As a radiation sub-model, this non-gray radiative property model can significantly improve prediction accuracy of radiative heat transfer in oxy-fuel combustion.     

气粒混合物非灰辐射特性合并宽窄谱带K分布模型

气粒混合物辐射问题具有全场性、非灰性、耦合性等特点,准确预估高温燃气/粒子非灰辐射特性是非常重要的。本文将合并宽窄谱带K分布模础(CWNBCK)与离散坐标法(DOM)结合,开展了非灰气粒混合物辐射换热问题的模拟工作,分别验证了一维和三维情况下应用该模型的准确性,给出不同工况下的热流源项、壁面热流或辐射热流等。结果表明:该模型能够给出与SNB模型精度基本相同的结果,考虑其计算效率的提高,可以在工程实际中应用该模型计算非灰气粒混合物辐射换热。