Multifractal Cascade Dynamics and Turbulent Intermittency

TLDR

Turbulent intermittency is central to many fields, and multifractal analysis is widely accepted as an effective way to quantify it. The authors contend that cascade processes are the essential physical models required for dynamical modeling of turbulent intermittency. They review recent advances that address the static, discrete, acausal, phenomenological shortcomings of existing cascade models, discuss universality classes, and introduce a continuous, space‑time anisotropic causal stochastic framework and the Scaling Gyroscope Cascade to link cascades with Navier–Stokes dynamics. Empirical analysis of velocity and temperature data confirms the strong universality model while rejecting the weak log‑Poisson alternative, demonstrates the predictability of intermittency fields via the causal stochastic model, and shows that the Scaling Gyroscope Cascade aligns universal parameters with empirical estimates, suggesting a viable path toward analytical renormalized intermittency models.

Abstract

Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we then show how to produce a causal stochastic model of intermittent fields and use it to study the predictability of these fields. Finally, by returning to the origins of the turbulent "shell models" and restoring a large number of degrees of freedom (the Scaling Gyroscope Cascade, SGC models) we partially close the gap between the cascades and the dynamical Navier–Stokes equations. Furthermore, we point out that beyond a close agreement between universal parameters of the different modeling approaches and the empirical estimates in turbulence, there is a rather common structure involving both a "renormalized viscosity" and a "renormalized forcing". We conclude that this gives credence to the possibility of deriving analytical/renormalized models of intermittency built on this structure.