Interior radiances in optically deep absorbing media—II Rayleigh scattering

Abstract

The interior radiances are calculated within an optically deep absorbing medium scattering according to the Rayleigh phase function. The accuracy of the matrix operator method is improved by many orders of magnitude through the use of accurate starting values obtained by the Runge-Kutta method rather than from the single scattering approximation. The radiance and flux are given for a range of solar zenith angles and for single scattering albedos of 1, 0.99, 0.9, 0.5 and 0.1. The development of the asymptotic angular distribution of the radiance is illustrated. It is shown that this asymptotic distribution is probably physically unobservable when ω 0 < 0.8, since the flux is less than 10 -8 of its original value at the beginning of the asymptotic region. The ratio of the upward to downward flux is calculated and is shown to be remarkably constant within the medium except very close to the boundaries. The heating rate within the medium is found to be very nearly proportional to the downward flux, except near the boundaries. When the single scattering albedo is small, a number of examples illustrate the significant contribution of the direct solar flux to the total flux even at great optical depths within the medium. The total downward flux decreases exponentially with optical depth away from boundaries when the single scattering albedo is greater than or equal to 0.9; when it is less than or equal to 0.5 only an approximate exponential fit can be obtained within the region accessible to experimental observation.