A Century of Turbulent Cascades and the Emergence of Multifractal Operators. Issue 3 (13th March 2020) - Schertzer, Daniel; Tchiguirinskaia, Ioulia
Abstract: A century of cascades and three decades of multifractals have built up a truly interdisciplinary framework that has enabled a new approach and understanding of nonlinear phenomena, in particular, in geophysics. Nevertheless, there seems to be a profound gap between the potentials of multifractals and their actual use. For instance, it seems ironic that multifractals have been mostly restricted to scalar‐valued fields, whereas cascades were first invoked for the wind velocity. We argue that this requires to proceed to new developments of the multifractal formalism and to the emergence of multifractal operators. This paper therefore aims to first simplify the introduction to the most recent developments based on the analysis and generation of multifractal fields with the help of the group property of the responses of a nonlinear system to a scale change. The generators of the multifractal operators are introduced with the help of symmetries as simple and basic as orthogonal rotations and mirror symmetries. This leads in a rather straightforward manner to the large class of Gauss–Clifford and Lévy–Clifford generators that combine a number of seductive properties, including universal statistical and robust algebraic properties. At the same time, we obtain new results on the entanglement of spherical and hyperbolic geometries, as well as on the existence of finite statistics of these cascades. Plain Language Summary: It is already exceptional that a quatrain has been inspiring theoretical and empirical research for a century, furthermore on a fundamental question of mathematical physics. It is moreover ironic that we are only now becoming able to address the real content of this quatrain. But that is what this paper is about! The fundamental question is that the basic properties of the solutions of the fundamental equations of fluid mechanics remain unknown, although these equations were derived almost two centuries ago. Therefore, the observed, extreme variability of the wind velocity still remains a puzzle. The aforementioned quatrain pointed out a cascade mechanism of transfer of the velocity from large to small wind structures. The major step discussed in the paper is to elucidate this transfer for vector quantities like wind, which have directions as well as magnitude, whereas cascades have been developed so far for scalar quantities that have only magnitude and no direction. This step should have many far‐reaching, theoretical and practical impacts due the importance of vector quantities, in particular, the wind velocity. Furthermore, it should help us to answer to the longstanding, aforementioned fundamental question of mathematical physics. Key Points: The concept of cascades has been inspiring for nonlinear geophysics for a century and has led to the breakthrough of multifractals New developments on stochastic algebra of cascade generators were needed for vectors to bridge the gap between potentials and applications Gauss‐Clifford and Lévy–Clifford generators combine statistical universality and robust algebraic properties