C.F. Segatto

The LTSN particular solution in a slab for an arbitrary source and large order of quadrature

Abstract In this work we split the homogeneous and particular LTSN solution taking advantage of the invariance property of the discrete ordinates directions, in order to handle transport problems with large N (1000). We present numerical simulations.

A General Analytical Solution of the Advection–Diffusion Equation for Fickian Closure

In the last few years there has been increased research interest in searching for analytical solutions for the advection–diffusion equation (ADE). By analytical we mean that no approximation is done along the derivation of the solution. There exists a significant literature regarding this theme. For illustration we mention the works of (Rounds 1955; Smith 1957; Scriven, Fisher 1975; Demuth 1978; van Ulden 1978; Nieuwstadt, de Haan 1981; Tagliazucca et al. 1985; Tirabassi 1989; Tirabassi, Rizza 1994; Sharan et al. 1996; Lin, Hildemann 1997; Tirabassi 2003). We note that in these works all solutions are valid for very specialized problems having specific wind and eddy diffusivities vertical profiles. Further, also in the literature there is the ADMM (Advection Diffusion Multilayer Method) approach which solves the two-dimensional ADE with variable wind profile and eddy diffusivity coefficient (Moreira et al. 2006). The main idea relies on the discretization of the Atmospheric Boundary Layer (ABL) in a multilayer domain, assuming in each layer that the eddy diffusivity and wind profile take averaged values. The resulting advection–diffusion equation in each layer is then solved by the Laplace transformation technique. For more details about this methodology see the review work done by (Moreira et al. 2006). We are also aware of the recent work of (Costa et al. 2006), dubbed as GIADMT method (Generalized Integral Advection Diffusion Multilayer Technique), which presented a general solution for the time-dependent three-dimensional ADE, again assuming the stepwise approximation for the eddy diffusivity coefficient and wind profile and proceeding further in similar way according the previous work. To avoid this approximation, in this work we report an analytical general solution for this problem, assuming that the eddy diffusivity coefficient and wind profile are arbitrary functions having a continuous dependence on the vertical and longitudinal variables. Without losing generality we specialize the application in micrometeorology, specially for the problem of simulation of contaminant releasing in the ABL.

General solution of one-dimensional approximations to the transport equation

Abstract The aim of this work is to present a complete review of the generic method for solving analytically one-dimensional approximations of the transport equation that appear as a set of first order linear differential equations, employing the Laplace transfrom technique over a finite domain. Recent advances are also included.

The determination of radiant parameters by the LTSN method

Abstract The aim of this work consists in the evaluation of the radiant parameters, transmissivity and reflectivity, in a heterogeneous plane parallel slab, for an incident radiation by the LTS N method. Numerical simulations are presented.

Solutions to the multidimensional linear transport equation by the spectral method

The spectral method is used to develop a solution for multidimensional transport problems for neutral particles in cartesian geometry. The procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a set of one-dimensional problems, whose solutions are well established. The convergence of this approach is studied in the context of the multidimensional discrete-ordinates equations.

Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium

Radiative transfer considers problems that involve the physical phenomenon of energy transfer by radiation in media. These phenomena occur in a variety of realms (Ahmad & Deering, 1992; Tsai & Ozisik, 1989; Wilson & Sen, 1986; Yi et al., 1996) including optics (Liu et al., 2006), astrophysics (Pinte et al., 2009), atmospheric science (Thomas & Stamnes, 2002), remote sensing (Shabanov et al., 2007) and engineering applications like heat transport by radiation (Brewster, 1992) for instance or radiative transfer laser applications (Kim & Guo, 2004). Furthermore, applications to other media such as biological tissue, powders, paints among others may be found in the literature (see ref. (Yang & Kruse, 2004) and references therein). Although radiation in its basic form is understood as a photon flux that requires a stochastic approach taking into account local microscopic interactions of a photon ensemble with some target particles like atoms, molecules, or effective micro-particles such as impurities, this scenario may be conveniently modelled by a radiation field, i.e. a radiation intensity, in a continuous medium where a microscopic structure is hidden in effective model parameters, to be specified later. The propagation of radiation through a homogeneous or heterogeneous medium suffers changes by several isotropic or non-isotropic processes like absorption, emission and scattering, respectively, that enter the mathematical approach in form of a non-linear radiative transfer equation. The non-linearity of the equation originates from a local thermal description using the Stefan-Boltzmann law that is related to heat transport by radiation which in turn is related to the radiation intensity and renders the radiative transfer problem a radiative-conductive one (Ozisik, 1973; Pomraning, 2005). Here, local thermal description means, that the domain where a temperature is attributed to, is sufficiently large in order to allow for the definition of a temperature, i.e. a local radiative equilibrium. The principal quantity of interest is the intensity I, that describes the radiation energy flow through an infinitesimal oriented area dΣ = ndΣ with outward normal vector n into the solid angle dΩ = ΩdΩ, where Ω represents the direction of the flow considered, with angle θ of the normal vector and the flow direction n · Ω = cos θ = μ. In the present case we focus on the non-linearity of the radiative-conductive transfer problem and therefore introduce the simplification of an integrated spectral intensity over all wavelengths or equivalently all frequencies that contribute to the radiation flow and further ignore possible effects due to polarization. Also possible effects that need in the formalism properties such as coherence 8

A parallel implementation of the LTSn method for a radiative transfer problem

A radiative transfer solver that implements the LTSn method was optimized and parallelized using the MPI message passing communication library. Timing and profiling information was obtained for the sequential code in order to identify performance bottlenecks. Performance tests were executed in a distributed memory parallel machine, a multicomputer based on IA-32 architecture. The radiative transfer equation was solved for a cloud test case to evaluate the parallel performance of the LTSn method. The LTSn code includes spatial discretization of the domain and Fourier decomposition of the radiances leading to independent azimuthal modes. This yields an independent radiative transfer equation for each mode that can be executed by a different processor in a parallel implementation. Speed-up results show that the parallel implementation is suitable for the used architecture.

On the convergence of the spherical harmonics approximations

Abstract The convergence of the spherical harmonics approximations using the isomorphism between two functional spaces and the approximation theorem of the C0-semigroup theory is proved.

On the Analytical Solution of the SN Radiative Transport Equation in a Slab for a Space-dependent Albedo Coefficient

In this work, we report a genuine general analytical solution for the linearized SN radiative-conductive transfer problem in a heterogeneous plane parallel atmosphere with the albedo coefficient depending continuously on the spatial variable. By general solution, we mean that the solution is valid for an arbitrary albedo coefficient continuous functions of the spatial variable having the property of fulfill the requirements of existence and uniqueness. The key feature of this novel approach embodies the steps: following the idea of the Decomposition method, we transform the original problem into a set of recursive problems with constant albedo coefficients, having the main feature that the sources terms takes the information of the spatial dependency of the albedo coefficient into account. This procedure allows us to solve, analytically, the resulting recursive system by the LTSN method developed for a constant albedo coefficient. Finally, we present the error control analysis of the solution and numerical comparisons against the literature results.

Existence Theory for the Solution of a Stationary Nonlinear Conductive‐Radiative Heat‐Transfer Problem in Three Space Dimensions

Abstract In this work we show that a stationary nonlinear coupled radiative‐conductive heat‐transfer problem in a convex bounded region with piecewise differentiable boundary in three dimensions under fairly general boundary conditions has a unique solution in a segment of a positive cone in a certain function space.