Functional renormalisation group for turbulence

Abstract

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying Navier-Stokes equations have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purpose, a sustained effort has been devoted to obtaining some effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the Renormalisation Group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier-Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, have been missing, which has thwarted for long RG applications. This framework is provided by the modern formulation of the RG called functional renormalisation group. The use of the FRG has rooted important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier-Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this {\it JFM Perspectives} a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We then show that the FRG enables one to describe turbulence forced at large scale, which was not accessible by perturbative means. We expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.