Universite Pierre et Marie Curie
Paris, France
Recent systematic satellite studies (LANDSAT, AVHRR, METEOSAT) of cloud radiances using (isotropic) energy spectra have displayed excellent scaling from at least about 300m to about 4000km (Figure 11.1), even for individual cloud pictures (Lovejoy et al., 1992). At first sight, this contradicts the observed diversity of cloud morphology, texture and type. Lovejoy and Schertzer (L&S) argue that the explanation of this apparent paradox is that the differences are due to anisotropy, e.g. differential stratification and rotation. A general framework for anisotropic scaling expressed in terms of isotropic self-similar scaling and fractals and multifractals is needed. Schertzer and Lovejoy (1985, 1991) have proposed Generalized Scale Invariance (GSI) in response to this need. In GSI, the statistics of the large and small scales of system can be related to each other by a scale changing operator Tl which depends only on the scale ratio l; there is no characteristic size.
From this definition it can be concluded that Tl must obey group properties: Tl=l- G where G is the generator. L&S showed by multifractal simulation how different cloud types and textures can be simulated by varying G. Similarly, they showed how to estimate G from satellite pictures.
L&S then turned to the problem of radiative transfer in fractal clouds (until 1990, with P. Gabriel, A. Davis, and G. L. Austin) and in multifractal clouds (since 1990). The most significant result of the research on monofractal clouds (occupying only a fractal subset of the space) is that two fundamental limits exist: the optically thick and optically thin cases. The bulk transmission and reflection properties, considering periodic boundary conditions and conservative scattering vary for thick clouds as T=ht-n, R=1-T, where n=1 in plane parallel homogeneous clouds but n<1 for the fractals. Here, h is a phase function. This means that for sufficiently thick clouds, plane parallel predictions can be seriously inaccurate.
Numerical results from single as well as multiple realizations underscore the dangers and large errors arising from invoking one-to-one relations between cloud parameters (e.g., large scale mean liquid water content) and large scale mean fluxes.
Finally, L&S outlined some recent results of scattering statistics on multifractal clouds (with B. Watson and G. Brosmalen), pointing out the fundamental qualitative difference between clouds with many and few sparse low-density regions; corresponding to the existence (or nonexistence, respectively) of negative statistical moments. Detailed study of the lognormal multifractals allowed L&S to develop a thick cloud formalism for photon transmission statistics (Lovejoy et al., 1992). Due to the near linearity of the photon path moment scaling function, renormalization of the optical density to an "equivalent" plane parallel density was a good approximation in this case. It gave very close agreement with direct simulations on thick lognormal multifractals. It also gave the same result as for the theoretical (one- dimensional) diffusion on lognormal multifractals (joint research with P. Silas). These stochastic radiative transfer results can explain the success of first- order Markov approximations which ignore high-order correlations in scatterings.
References
Lovejoy, S., D. Schertzer, B. Watson, 1992: Radiative Transfer and Multifractal Clouds: theory and applications, I.R.S., 92, A. Arkin et al., Eds. 108-111.
Schertzer, D., S. Lovejoy, 1985: Generalized scale invariance in turbulent phenomena, Physico-Chemical Hydrodynamics Journal, 6, 623-635.
Schertzer, D., S. Lovejoy, 1991: Nonlinear geodynamical variability: Multiple singularities, universality and observables. Scaling, fractals and non-linear variability in geophysics, D. Schertzer, S. Lovejoy, Eds., 41-82, Kluwer.
