Generalization of the spherical harmonic method to radiative transfer in multidimensional space.
Abstract
The basic radiative transfer equation in threedimensional space is expressed in terms of three commonly used coordinate systems, namely, Cartesian, cylindrical and spherical coordinates. The concept of a transformation matrix is applied to the transformation processes between the Cartesian system and two other systems. The spherical harmonic method is then applied to decompose the radiative transfer equation into a set of coupled partial differential equations for all three systems in terms of partial differential operators. By truncating the number of partial differential equations into four along with further mathematical analysis, we obtain a modified Helmholtz equation. For each coordinate system, analytical solutions in terms of infinite series are obtained whenever the equation is solvable by the technique of separation of variables with proper boundary conditions. Numerical computations are carried out for one dimensional radiative transfer to illustrate the applicability of the technique developed in the present study.
 Publication:

Journal of Quantitative Spectroscopy and Radiative Transfer
 Pub Date:
 October 1982
 DOI:
 10.1016/00224073(82)900280
 Bibcode:
 1982JQSRT..28..271O
 Keywords:

 Cloud Physics;
 Radiative Transfer;
 Spherical Harmonics;
 Approximation;
 Cartesian Coordinates;
 Partial Differential Equations;
 Spherical Coordinates;
 Physics (General);
 Radiative Transfer