Single scattering phase function

Up to this point, exact analytic expressions for thin layers have been developed, as well as numerical procedures for extending these solutions to thick layers. A straightforward procedure for solving the transfer equation for thick nonscattering atmospheres has been developed. It remains to find a satisfactory approximate solution for thick scattering atmospheres, since exact analytic solutions do not exist. 

The two-stream approximation is just such a solution. The continuous radiation field is replaced by two directed beams, one up and one down. Because the solution is analytic, it is helpful to separate the azimuthal and polar angles. We write the phase function [cf. Eq. (2.1.3)] as a finite series of Legendre polynomials  

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The problem is cast in spherical coordinates in order to ensure the surface of the particle coincides with one of the coordinate surfaces, making it much easier to impose the boundary conditions correctly. Solutions to the scalar wave equation are then used to derive two vector fields, linear combinations of which satisfy Maxwell s equations. Particular integrals are found by requiring continuity of the field components across the particle surface. The scattered field at large distances from the particle is then evaluated, leading to explicit expressions for particle cross sections and the single scattering phase function. 

The radial components (Eor, 7/or) are of order (1/r ) as r -> oo, and thus do not contribute that is, the scattered wave becomes transverse at large r. In order to obtain the single-scattering phase function we are interested only in relative fluxes, and can set the flux equal to the square of the real amplitude of the electric vector. The two polarization components of the flux are... 

The results of multiple scattering calculations based on computed single scattering phase functions of selected types of crystals compare well with observed fluxes in some cases [71, 72, 73] however, in general the microphysical parameters of ice clouds cannot be measured with the accuracy needed for climate studies [76]. Puzzles remain as to the very nature of certain radiatively important ice clouds namely, subvisual cirrus ( e.g., [77]), clouds of optical depth less than 0.03 in the upper troposphere, and diamond dust 10-50 ixm facetted crystals and clear sky ice precipitation , irregularly shaped frozen particles in the lower Arctic cloudfree sky [78]. 

This approximation divides the directional distribution of scattered energy into two components one highly peaked component in the forward direction, and an isotropic distribution for all other directions. Improvement of this phase function is possible if the second term on the right-hand side of Eq. 7.102 is comprised of more than a single term. Crosbie and David-... 

Let us suppose now, that the effective single-scattering albedo of the modified phase matrix (m, v) has been reduced substantially by means of an appropriate choice of the free parameter vectors in Eq. (36). Then, with increasing optical depth, the surface Green s function matrix G (t, m 0, /Iq ) rapidly fades away, and Eq. (45) asymptotically yields... 

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