Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems

Scaling is an important measure of multi-scale fluctuation systems. Turbulence as the most remarkable multi-scale system possesses scaling over a wide range of scales. She-Leveque (SL) hierarchical symmetry, since its publication in 1994, has received wide attention. A number of experimental, numerical and theoretical work have been devoted to its verification, extension, and modification. Application to the understanding of magnetohydrodynamic turbulence, motions of cosmic baryon fluids, cosmological supersonic turbulence, natural image, spiral turbulent patterns, DNA anomalous composition, human heart variability are just a few among the most successful examples. A number of modified scaling laws have been derived in the framework of the hierarchical symmetry, and the SL model parameters are found to reveal both the organizational order of the whole system and the properties of the most significant fluctuation structures. A partial set of work related to these studies are reviewed. Particular emphasis is placed on the nature of the hierarchical symmetry. It is suggested that the SL hierarchical symmetry is a new form of the self-organization principle for multi-scale fluctuation systems, and can be employed as a standard analysis tool in the general multi-scale methodology. It is further suggested that the SL hierarchical symmetry implies the existence of a turbulence ensemble. It is speculated that the search for defining the turbulence ensemble might open a new way for deriving statistical closure equations for turbulence and other multi-scale fluctuation systems. The project supported by the National Natural Science Foundation (90716008) and by MOST 973 project (2009CB724100).