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)'.