Functional renormalisation group approach to shell models of turbulence

TL;DR

This paper develops a functional renormalisation group framework for shell models of turbulence, revealing how different forcing types lead to distinct universality classes and scaling behaviors.

Contribution

It introduces an inverse RG flow formalism to analyze shell models, highlighting the impact of forcing types on turbulence universality classes.

Findings

Power-law forcing results in dimensional (K41-like) scaling.

Large-scale forcing leads to anomalous scaling.

Different forcing types define distinct fixed points and universality classes.

Abstract

Shell models are simplified models of hydrodynamic turbulence, retaining only some essential features of the original equations, such as the non-linearity, symmetries and quadratic invariants. Yet, they were shown to reproduce the most salient properties of developed turbulence, in particular universal statistics and multi-scaling. We set up the functional renormalisation group (RG) formalism to study generic shell models. In particular, we formulate an inverse RG flow, which consists in integrating out fluctuation modes from the large scales (small wavenumbers) to the small scales (large wavenumbers), which is physically grounded and has long been advocated in the context of turbulence. Focusing on the Sabra shell model, we study the effect of both a large-scale forcing, and a power-law forcing exerted at all scales. We show that these two types of forcing yield different fixed points, and thus correspond to distinct universality classes, characterised by different scaling exponents. We find that the power-law forcing leads to dimensional (K41-like) scaling, while the large-scale forcing entails anomalous scaling.