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Universal Law of Cascade of Turbulent Fluctuations

In turbulence, fluctuations are developed over a wide range of scales and amplitudes as a result of strong nonlinear interactions. As fluctuations cascade, both mechanical law and statistical law are important to determine the overall structures. The mechanical law manifests in the development of the so-called most intermittent structures, while the statistical law expresses itself in terms of a certain symmetry leading to the organization of the fluctuations into a whole hierarchy. This view of the Hierarchical Structure model (She and Leveque, Phys. Rev. Lett. 72 (1994), 366) of turbulence is discussed in detail here with some recent analytical development. It is hoped that the model provides inspirations for a renewed phenomenology of turbulence guiding the development of a more complete theory of turbulence.