Over the last twenty years, many studies have been made of radiative transfer in scaling cloud fields, the vast majority of which have been limited to numerical studies in clouds with relatively small optical thickness. An exception to this was the development of a formalism for treating single scattering in optically thick but conservative multifractal clouds without significant holes. In part I of this paper we show how these results can be extended to general “universal” multifractal clouds dominated by low density “Lévy holes”. In part II, we demonstrate how the analytic single scattering results can be generalized to multiple scattering including the case of very thick clouds as well as to realistic nonconservative clouds.

In Part I of this paper, we developed asymptotic approximations for single photon scattering in thick, highly heterogeneous, “Log-Lévy” multifractal clouds. In Part II, theoretical multiple scattering predictions are numerically tested using Monte Carlo techniques, which show that, due to long range correlations, the photon paths are “subdiffusive” with the corresponding fractal dimensions tending to increase slowly with mean optical thickness. We develop reasonably accurate statistical relations between N scatter statistics in thick clouds and single scatter statistics in thin clouds. This is explored further using discrete angle radiative transfer (DART) approach in which the radiances decouple into non-interacting families with only four (for 2-D clouds) radiance directions each. Sparse matrix techniques allow for rapid and extremely accurate solutions for the transfer; the accuracy is only limited by the spatial discretization.By “renormalizing” the cloud density, we relate the mean transmission statistics to those of an equivalent homogeneous cloud. This simple idea is remarkably effective because two complicating effects act in contrary directions: the “holes” which lead to increased single scatter transmission and the tendency for multiply scattered photons to become “trapped” in optically dense regions, thus decreasing the overall transmission.

Theories of atmospheric dynamics, turbulence, cloud structure and atmospheric radiative transfer are nearly all based on the idea that at small enough scales, structures are three dimensional whereas, at large scales they are two dimensional. Clouds can readily be hundreds or thousands of kilometers long, however due to gravity, they form layers which are not nearly so thick: when viewed from space the entire atmosphere appears as an envelope roughly 10 km thick surrounding the Earth like an onion skin. This has lead to the popular idea that the large scale structures-the "weather"-is more nearly two dimensional; theoretically it is justified by the "quasi-geostrophic" approximation.