RG approach to the inviscid limit for shell models of turbulence

Abstract

We consider an initial value problem for shell models that mimic turbulent velocity fluctuations over a geometric sequence of scales. Our goal is to study the convergence of solutions in the inviscid (more generally, vanishing regularization) limit and explain the universality of both the limiting solutions and the convergence process. We develop a renormalization group (RG) formalism representing this limit as dynamics in a space of flow maps. For the dyadic shell model, the RG dynamics has a fixed-point attractor, which determines universal limiting solutions. Deviations from the limiting solutions are also universal and given by a leading eigenmode (eigenvalue and eigenvector) of the linearized RG operator. Application to the Gledzer shell model reveals the RG attractor in the form of a closed invariant curve, while the Sabra shell model yields chaotic RG dynamics. An important consequence of the RG formalism is the understanding of the different roles of symmetry-preserving (canonical) and symmetry-breaking (e.g. viscous) regularizations.