Multifractal techniques and notions are increasingly widely recognized as the most appropriate and straightforward framework within which to analyze and simulate not only the scale dependence of geophysical observables, but also their extreme variability over a wide range of scales. This is particularly the case for cloud fields and their radiative properties. Schertzer first recalled the original scalar framework of turbulent cascades, especially for the modeling and analysis of passive clouds, based on multifractal developments of the Corrsin-Obukhov spectral scaling of scalar variance (Schertzer and Lovejoy, 1987; Pecknold et al. 1993). These developments are based on the scaling symmetries of the dynamical equations of both the velocity and liquid water density fields. He emphasized the power of straightforward simulation methods based on these physical arguments. Schertzer showed a video (Brenier et al. 1990) displaying a time evolution of multifractal cloud in the framework of universal multifractals. He insisted that with the aid of these tools, there is no real need to look for constructs such as bounded cascades.
There are two rather recent developments in stochastic multifractals that are of particular interest. On the one hand, as a wide variety of possible multifractal behaviors exists, ranging from extremely "soft" to "hard," there is a need to understand their differences qualitatively, as well as the transitions from one type of behavior to another. Different behaviors correspond to multifractal analogues of phases, with two types of phase transitions between them. High- or low-temperature second-order transitions naturally arise from finite sample sizes and are only representative of these limitations. By contrast, low-temperature first-order transitions are consequences of the scale and dimension of the observations, which are no longer able to smooth away the most extreme small-scale fluctuations by building up larger-scale structures. The latter is a generic stochastic route to a non-classical self-organized criticality, since it occurs in high-dimensional stochastic multifractal processes with non- vanishing input (e.g. flux of turbulent energy). The origin of these transitions, as well as their implications, were discussed. There are many practical and drastic consequences of the "divergence of moments" associated with self-organized criticality, e.g. the breakdown of laws of large numbers and the related standard statistical estimates, the loss of ergodicity, the existence of very large fluctuations related to the "zoology" of structures, and the overwhelming contribution of catastrophic events.
On the other hand, the present situation is somewhat paradoxical: classical methods, such as those used in GCM research or direct simulations, deal easily and explicitly with this vector interaction, but only over a very limited range of scales, whereas scaling models deal easily with an infinite range of scales but avoid treating this vector interaction. For instance, multifractal modeling of clouds has relied until now on the simplifying hypothesis that the interaction between the cloud and the environmental dynamics can be reduced to a scalar relationship (namely between their respective fluxes). On the theoretical side, there now exists a rather general framework of "Lie cascades" which has been recently developed to analyze and generate multiplicative processes for vector and tensor fields, and more generally rather abstract fields admitting a Lie group of symmetries. This framework opens up a very appealing alternative to GCM techniques, since we then may consider the generator of the (scaling) multi-component field describing atmospheric states (e.g. dynamics, temperature, water concentration, radiative fields, etc.).
In conclusion, Schertzer briefly emphasized that these aspects of new developments of multifractals are potentially very powerful techniques which may allow us to simulate and analyze both qualitatively and quantitatively a wide variety of geophysical fields and interactions, well beyond conventional deterministic frameworks. In the future, these tools should be useful for evaluating and assessing many aspects of global change.
References
Brenier, P., D. Schertzer, A. Davis, D. Lavalle, S. Lovejoy, J. Wilson, 1991: Multifractal Dynamics. Video distributed by World Scientific, Singapore.
Pecknold, S., S. Lovejoy, D. Schertzer, C. Hooge, J. F. Malouin, 1993: The simulation of universal multifractals. Cellular Automata: prospects in astronomy and astrophysics, Eds. J. M. Perdang, A. Lejeune, World Scientific, 228-267.
Schertzer, D., S. Lovejoy, 1987: Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades. J. Geophys. Res., 92, 9692-9714.
Schertzer, D., S. Lovejoy, 1991: Scaling, Fractals and Non-Linear Variability in Geophysics, Kluwer, Dordrecht-Boston, 318 pp.
Schertzer D. and S. Lovejoy, 1994: Non Linear Variability in Geophysics 3 (NVAG3) Lectures Notes: Scaling and Multifractal processes. World Scientific, Singapore, 292 pp. (in press)