Approximate polarization calculations with scalar radiative transfer theory

The Pomraning phase function can be used to perform approximate polarized Rayleigh transfer calculations with a scalar radiative transfer equation. The approximation is numerically tested for the albedo problem consisting of azimuthally independent radiation incident on a homogeneous semi-infinite atmosphere. The numerical tests were carried out with the same approach used by Viik (JQSRT 68 (2000) 319-326) to numerically test the approximate phase function for solving the Milne problem. Away from the surface the Pomraning phase function gives marginally better results for the diffuse radiation than the usual scalar Rayleigh phase function because it was derived from an asymptotic limit more appropriate for deeper locations in an atmosphere. For optical depths less than unity, though, the scalar Rayleigh approximation is better than the Pomraning approximation.