The vast majority of treatments of radiative transfer inheterogeneous clouds have been limited to numerical studies in clouds which are relatively optically thin. An exception to this was the development of a formalism for treating single scattering in optically thick but special (conservative) multifractal clouds [Lovejoy, et al., 1995] (see fig. 1). In this presentation we show how these results can be extended to nonconservative general universal multifractal clouds, and how the analytic single scattering results can be generalized to multiple scattering at least for optical thickness below a super thick limit (?>>100). The theoretical analytic results thus give accurate predictions for the mean cloud optical properties of clouds with realistic multifractal parameters and cloud thicknesses; for example using the observed multifractal cloud characteristics, we predict that the mean cloud transmission decreases with the 0.88 power of the optical thickness (the homogeneous exponent is unity). For clouds with mean optical thickness of 100 (with 1-g=0.15); this is a 38% effect with respect to homogeneity. These theoretical multiple scattering predictions are numerically tested using the 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. Since in thick clouds the phase function is of secondary importance (it doesn't affect the scaling exponents), this approach is justified in optically thick clouds where photons undergo many scatterings; sparse matrix techniques allow for rapid and extremely accurate solutions for the transfer.
By varying the extinction coefficient, we were able to study the effect of increasing cloud thickness, for typical cloud mean optical thickness in the range 8-200. In this paper we present new results on the transmission statistics in general 2-D universal multifractal clouds (for various Hurst exponents and levels of intermittency). By renormalizing the radiation, we relate the mean transmission statistics to those of a homogeneous cloud. Preliminary results concerning the effects of stratification and more generally of anisotropy (associated with different cloud types) are also presented.
Fig. 1: These 3D simulations respect all the observed anisotropic, turbulent statistics in the vertical and horizontal. The different realizations have different scaling anisotropies corresponding to different cloud types. Only the single scattered visible radiation field is shown, the radiation field is from above with uniform surface below, radiation in a single direction is shown (the clouds are effectively viewed from infinity). More details may be found at the multifractal explorer site:
Lilley, M., et al. (2004), 23/9 dimensional anisotropic scaling of passive admixtures using lidar aerosol data, Phys. Rev. E, 70, 036307-036301-036307.
Lovejoy, S., et al. (1995), Scattering in multifractal media., Proceedings Particle Transport in Stochastic Media, Ed. L. Briggs, American Nuclear Society, Portland, Or., p750-760.
Lovejoy, S., and D. Schertzer (2005), Multifractals, cloud radiances and rain, J. Hydrol., (in press).
Lovejoy, S., Schertzer, D., Stanway, J. D. (2001), Direct Evidence of planetary scale atmospheric cascade dynamics, Phys. Rev. Lett., 86, 5200-5203.