Abstract
This note is devoted to broken and emerging scale invariance of turbulence. Pumping breaks the symmetry: the statistics of every mode explicitly depend on the distance from the pumping. And yet the ratios of mode amplitudes, called Kolmogorov multipliers, are known to approach scale-invariant statistics away from the pumping. This emergent scale invariance deserves an explanation and a detailed study. We put forward the hypothesis that the invariance of multipliers is due to an extreme non-locality of their interactions (similar to the appearance of mean-field properties in the thermodynamic limit for systems with long-range interaction). We analyse this phenomenon in a family of models that connects two very different classes of systems: resonantly interacting waves and wave-free incompressible flows. The connection is algebraic and turns into an identity for properly discretized models. We show that this family provides a unique opportunity for an analytic (perturbative) study of emerging scale invariance in a system with strong interactions.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.
Highlights
We owe to Uriel Frisch that scale invariance and its breakdown came into focus for physicists and mathematicians working on turbulence [1]
This note weaves together three threads introduced by Uriel Frisch into turbulence studies: criterion for cascade direction, breakdown of self-similarity, and multi-fractal formalism
That possibility is based on the remarkable fact that thermal equilibrium has exactly Gaussian statistics with independent modes; varying the parameter of our family, one is able to study turbulence close to thermal equilibrium
Summary
We owe to Uriel Frisch that scale invariance and its breakdown came into focus for physicists and mathematicians working on turbulence [1]. As far as scale invariance is concerned, statistics of the velocity gets independent of the (long) distance to the dissipation scale, but changes with the distance to the pumping scale no matter how large this distance is Understanding this phenomenon required introduction of simple yet non-trivial models [3] going back to Kolmogorov and developed by Obukhov, Frisch, Kraichnan and others. The distinction between direct and inverse cascades was noticed: the former had an anomalous scaling, while the latter had not [5,6] Another aspect of the symmetry breakdown was related to the fractal distribution of the dissipation in space [1]. We address the broken and emerging scale invariance of turbulent cascades in a class of systems capable to model two distinct classes of physical phenomena. We shall consider one family of such models and describe far-from-equilibrium (turbulent) states of such systems
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