A closed-form solution of differential approximation for radiative transfer in a planar refractive medium

The closed-form solution of differential approximation (DA) for radiative transfer in a planar, scattering, refractive medium can be derived when the medium is in radiative equilibrium or purely scattering. In this work, linearly and exponentially spatial variations of refractive index are considered. The results obtained by the DA agree well with those obtained by the discrete ordinates method (DOM) and the Monte Carlo method (MCM) for the optically thick cases and the discrepancy between the DA results and the accurate numerical solutions obtained by the DOM and the MCM increases as the optical thickness decreases. The discrepancy between the results obtained by the DA and by the DOM also increases as the gradient of the linear refractive index increases. Besides, the influence of the scattering albedo and the scattering phase function coefficient on the accuracy of the analytical DA solution is less noticeable. Since the internal reflection due to the exponential decay of refractive index increases with the variation range of refractive index, the dimensionless radiative flux decreases with the increase of variation range of refractive index in a purely scattering medium. The present results show that the effect of the internal reflection is stronger in a thinner medium with an exponentially decaying refractive index.