Emerging scale invariance in a model of turbulence of vortices and waves

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)'.

Automated summary

This note discusses the broken and emerging scale invariance of turbulence, focusing on the non-local interactions that may lead to scale-invariant statistics of mode amplitudes known as Kolmogorov multipliers. The hypothesis put forward suggests that extreme non-locality of interactions plays a crucial role in the invariance of multipliers. This study provides a unique opportunity for an analytical investigation of emerging scale invariance in systems with strong interactions.