In the first two installments of this series, various cloud models were studied with angularly discretized versions of radiative transfer. This simplification allows the effects of cloud inhomogeneity to be studied in some detail. The families of scattering media investigated were those whose members are related to each other by scale changing operations that involve only ratios of their sizes (''scaling'' geometries). In part 1 it was argued that, in the case of conservative scattering, the reflection and transmission coefficients of these families should vary algebraically with cloud size in the asymptotically thick regime, thus allowing us to define scaling exponents and corresponding ''universality'' classes. In part 2 this was further justified (by using analytical renormalization methods) for homogeneous clouds in one, two, and three spatial dimensions (i.e., slabs, squares, or triangles and cubes, respectively) as well as for a simple deterministic fractal cloud. Here the same systems are studied numerically. The results confirm (1) that renormalization is qualitatively correct (while quantitatively poor), and (2) more importantly, they support the conjecture that the universality classes of discrete and continuous angle radiative transfer are generally identical.

Additional numerical results are obtained for a simple class of scale invariant (fractal) clouds that arises when modeling the concentration of cloud liquid water into ever smaller regions by advection in turbulent cascades. These so-called random ''β models'' are (also) characterized by a single fractal dimension. Both open and cyclical horizontal boundary conditions are considered. These and previous results are constrasted with plane-parallel predictions, and measures of systematic error are defined as ''packing factors'' which are found to diverge algebraically with average optical thickness and are significant even when the scaling behavior is very limited in range. Several meteorological consequences, especially concerning the ''albedo paradox'' and global climate models, are discussed, and future directions of investigation are outlined. Throughout this series it is shown that spatial variability of the optical density field (i.e., cloud geometry) determines the exponent of optical thickness (hence universality class), whereas changes in phase function can only affect the multiplicative prefactors. It is therefore argued that much more emphasis should be placed on modeling spatial inhomogeneity and investigating its radiative signature, even if this implies crude treatment of the angular aspect of the radiative transfer problem. ¿ American Geophysical Union 1990