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Functional renormalisation group for turbulence

Published online by Cambridge University Press:  24 October 2022

Léonie Canet*
Affiliation:
Université Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France Institut Universitaire de France, 1 rue Descartes, 75005 Paris, France
*
Email address for correspondence: leonie.canet@grenoble.cnrs.fr
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Abstract

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the 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 purposes, a sustained effort has been devoted to obtaining an 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, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled 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 JFM Perspectives article 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 stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, 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.

Type
JFM Perspectives
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1. Introduction

This JFM Perspectives article proposes a guided tour through the recent achievements and open prospects of the functional renormalisation group (FRG) formalism applied to the problem of turbulence. Why should the renormalisation group (RG) be useful at all to study turbulence? The very essence of the renormalisation group is to eliminate degrees of freedom in a systematic way, by averaging over fluctuations at all scales, starting from the fundamental description of a system. It thereby provides an effective model ‘from first principles.’ This is precisely what is needed, and often lacking, in many turbulence problems. Indeed, the fundamental equations for fluid dynamics are the Navier–Stokes (NS) equations, but the number of degrees of freedom necessary to describe turbulence grows as Re $^{9/4}$, where Re is the Reynolds number. Conceiving an appropriate modelling, e.g. devising some reliable effective equations to simplify the problem, is a major goal in many applications. This is of course a very difficult task in general, and the RG certainly offers a valuable tool, at least for idealised situations. The program of applying RG to turbulence started in the early eighties, but it was bound to rely on the tools available at that time, which were based on perturbative expansions. However, there is no small parameter for turbulence, and this program was largely thwarted. The only way out was to introduce in the theoretical description a forcing of the turbulence with a power-law spectrum, i.e. acting at all scales, which is unphysical (Eyink Reference Eyink1994). Results for a physical large-scale forcing are then extrapolated from a certain limit which cannot be justified. This is explained in more detail in § 2.

In the meantime, functional and non-perturbative ways of realising the RG procedure were developed (Wetterich Reference Wetterich1993; Morris Reference Morris1994). They allow one in particular to incorporate from the onset a physical large-scale forcing, and thereby to completely circumvent the inextricable issues posed by perturbative expansions. Indeed, the functional renormalisation group (FRG) offers a very powerful way to implement non-perturbative – yet controlled – approximation schemes, i.e. approximations not requiring the existence of a small parameter. The main scheme consists in the use of an ansatz for a central object in field theory which is called the effective action (see § 3). The construction of this ansatz relies on the fundamental symmetries of the problem. This approach has yielded many results, allowing one to obtain very accurate – although approximate – estimates for various quantities, including whole functions, not only numbers. We present as examples in § 5 the results for the scaling functions for the Burgers equation, and in § 6 the results for the energy spectrum, the second- and third-order structure functions for the NS equations, including an accurate estimate of the Kolmogorov constant. More strikingly, the FRG can also be used to obtain exact results, which are not based on an ansatz. In turbulence, these results are explicit expressions for multi-point Eulerian spatio-temporal correlation functions. They are obtained as an exact limit at large wavenumbers. Given the very few exact results available for three-dimensional (3-D) turbulence, this result is particularly remarkable and is the main topic of this Perspectives article.

Let us emphasise that all the results obtained so far with FRG concern stationary, homogeneous and isotropic turbulence. This Perspectives article is thus focused on this idealised situation. However, the scope of application of FRG is not restricted to this situation. It certainly cannot tackle arbitrarily complicated situations and would be of no use to determine the precise flow configuration around particular airfoils or in some specific meteorological conditions. However, it could be used, and be efficient, for small-scale modelling, as it is designed to provide an effective description, e.g. the form of the coarse-grained effective equations, at any given scale, in particular at the grid scale of large-scale numerical simulations for instance. This direction of research is still in its infancy, and will not be further developed in this contribution, although it definitely maps out a promising program for the future.

Let us also point out that there can be different levels of reading of this Perspectives article. Some parts (e.g. §§ 3, 4, 7.2 and 7.3) are rather technical, with the objective of providing a comprehensive introduction on the basic settings of the FRG for turbulence, and more importantly on the essential ingredients, in particular symmetries, that enter approximation schemes in this framework. Therefore, the underlying hypotheses are clearly stated to highlight the range of validity of the different results presented. Other parts (e.g. §§ 5.3, 5.4, 6.4, 8 and 9) are devoted to illustrating the physical implications of these results, by comparing them with actual data and observations from experiments and direct numerical simulations (DNS). Therefore one may first wish to focus on the concrete outcomes of FRG, deferring the technical aspects.

In detail, this JFM Perspectives article is organised as follows. We start in § 2 by stressing the interest of RG as a means to study turbulence and its different implementations. We explain in § 3.1 how one can represent stochastic fluctuations as a path integral, to arrive at the field theory for turbulence in § 3.2. A key advantage of this formulation is that it provides a framework to deeply exploit the symmetries. We show in § 3.3 how one can derive exact identities relating different correlation functions from symmetries (and extended symmetries) of the field theory. These identities include well-known relations such as the Kármán–Howarth or Yaglom relations (§ 3.4), but are far more general. The FRG formalism is introduced in § 4 with the standard approximation schemes used within this framework. To illustrate the application of this method, we start in § 5 with the Burgers equation as a warm-up example. We then present in § 6 a first important result for NS turbulence ensuing from FRG, which is the existence of a fixed point corresponding to the stationary turbulence forced at large scales, and the determination of the associated statistical properties. The main achievement following from FRG method, addressed in § 7, is the derivation of the formal general expression of multi-point spatio-temporal correlation functions at large wavenumbers. The rest of this Perspectives article is dedicated to illustrating these results, for NS turbulence in § 8, and for passive scalar turbulence in § 9. Another result, concerning the form of the kinetic energy spectrum in the near-dissipation range of scale is briefly mentioned in § 8.4. Finally, the conclusion and perspectives are gathered in § 10.

2. Scale invariance and the RG

2.1. Renormalisation group and turbulence

Let us explain why RG should be useful to study turbulence from the point of view of statistical physics. The aim of statistical physics is to determine from a fundamental theory at the microscopic scale the effective behaviour at the macroscopic scale of the system, comprising a large number ${\sim }N_A$ of particles (in a broad sense). This requires us to average over stochastic fluctuations (thermal, quantum, etc$\ldots$). When the fluctuations are Gaussian, and elementary constituents are non-interacting, central limit theorem applies and allows one to perform the averaging (which is how one obtains e.g. the equation of states of the perfect gas). However, when the system becomes strongly correlated, this procedure fails since the constituents are no longer statistically independent. This problem appeared to be particularly thorny for critical phenomena, and has impeded progress in our theoretical understanding for a long time. Indeed, at a second-order phase transition, the correlation length of the system diverges, which means that all the degrees of freedom are correlated and fluctuations develop at all scales. The divergence of the correlation length leads to the quintessential property of scale invariance, characterised by universal scaling laws, with anomalous exponents, i.e. exponents not complying with dimensional analysis.

The major breakthrough in understanding the physical origin of this anomalous behaviour, and primarily to compute it, was achieved with the RG. Although the RG had already been used under other forms in high-energy physics, it acquired its plain physical meaning from the work of K. Wilson (Wilson & Kogut Reference Wilson and Kogut1974). The RG provides the tool to perform the average over fluctuations, whatever their nature, i.e. eliminates degrees of freedom, even in the presence of strong correlations, and one is able thereby to build the effective theory at large scale from the microscopic one. One of the key features of the RG is that all the useful information is encoded in the RG flow, i.e. in the differential equation describing the change of the system under a change of scale. In particular, a critical point, associated with scale invariance, corresponds in essence to a fixed point of the RG flow. Let us notice that the notion of large scale is relative to the microscopic one, and it depends on the context. For turbulence, the microscopic scale, denoted $\varLambda ^{-1}$, refers to the scale at which the continuum description of a fluid, in terms of the NS equation, is valid, say the order of the mean free path in the fluid (much smaller than the Kolmogorov scale $\eta$). The ‘large’ scale of the RG then refers to typical scales at which the behaviour of the fluid is observed, i.e. inertial or dissipation ranges, thus including the usual ‘small’ scales of turbulence.

The analogy between critical phenomena and turbulence is obvious, and was early pointed out in Nelkin (Reference Nelkin1974), and later refined in e.g. (Eyink & Goldenfeld Reference Eyink and Goldenfeld1994). Indeed, when turbulence is fully developed, one observes in the inertial range universal behaviours, described by scaling laws with anomalous critical exponents, akin to an equilibrium second-order phase transition. As the RG had been so successful in the latter case, it early arose as the choice candidate to tackle the former. Concomitantly, the RG was extended to study not only the equilibrium but also the dynamics of systems (Martin, Siggia & Rose Reference Martin, Siggia and Rose1973; de Dominicis Reference de Dominicis1976; Janssen Reference Janssen1976), and the first implementations of the ‘dynamical RG’ to study turbulence date back to the early eighties (Forster, Nelson & Stephen Reference Forster, Nelson and Stephen1977; DeDominicis & Martin Reference DeDominicis and Martin1979; Fournier & Frisch Reference Fournier and Frisch1983; Yakhot & Orszag Reference Yakhot and Orszag1986). However, the formulation of the RG has remained intimately linked with perturbative expansions, relying on the existence of a small parameter. This small parameter is generically chosen as the distance $\varepsilon = d_c-d$ to an upper critical dimension $d_c$, which is the dimension where the fixed point associated with the phase transition under study becomes Gaussian, and the interaction coupling vanishes. In the paradigmatic example of the $\phi ^{4}$ theory which is the name of the field theory describing the second-order phase transition in the Ising universality class, the interaction coupling $g$ has a scaling dimension $L^{4-d}$. Thus it vanishes in the $L\to \infty$ limit for $d\geq d_c=4$, which means that fluctuations become negligible and the mean-field approximation suffices to provide a reliable description. The Wilson–Fisher fixed point describing the transition below $d_c$ can then be captured by a perturbative expansion in $\varepsilon = d_c-d$ and the coupling $g$.

In contrast, in turbulence, the ‘interaction’ is the nonlinear advection term, whose ‘coupling’ is unity, i.e. it is not small, and does not vanish in any dimension. Thus, one lacks a small parameter to control perturbative expansion. The usual strategy has been to introduce an artificial parameter $\varepsilon$ through a forcing with power-law correlations behaving in wavenumber space as $p^{4-d-\varepsilon }$, i.e. applying a forcing on all scales, which is unphysical (Eyink Reference Eyink1994). (Note that the letter $p$ is used to denote a wavenumber throughout this Perspectives article. The pressure is denoted with a letter ${\rm \pi}$ to avoid any confusion.) Fully developed turbulence in $d=3$ should then correspond to an infrared (IR) dominated spectrum of the stirring force, as occurs for $\varepsilon \geq 2$, hence for large values for which the extrapolation of the perturbative expansions is fragile. Moreover, one finds an $\varepsilon$-dependent fixed point, with an energy spectrum $E(p)\propto p^{1-2\varepsilon /3}$. The value corresponding to the Kolmogorov 1941 theory (denoted K41 in the following) is recovered for $\varepsilon =4$, but this value should somehow freeze for $\varepsilon$ larger than 4, and such a freezing mechanism can only be invoked within the perturbative analysis (Fournier & Frisch Reference Fournier and Frisch1983). In fact, the situation is even worse since it was recently shown that the turbulence generated by a power-law forcing or by a large-scale forcing is simply different (it corresponds to two distinct fixed points of the RG, in particular the latter is intermittent whereas the former is not for any value of $\varepsilon$ including $\varepsilon =4$) (Fontaine et al. Reference Fontaine, Tarpin, Bouchet and Canet2022). Therefore, NS turbulence with a large-scale forcing simply cannot be extrapolated from the setting with power-law forcing. Not only does recovering the K41 spectrum turn out to be difficult within this framework, but the first RG analyses also failed to capture the sweeping effect (Chen & Kraichnan Reference Chen and Kraichnan1989; Yakhot, Orszag & She Reference Yakhot, Orszag and She1989; Nelkin & Tabor Reference Nelkin and Tabor1990), and led to the conclusion that one should go to a quasi-Lagrangian framework to obtain a reliable description (L'vov & Procaccia Reference L'vov and Procaccia1995; L'vov, Podivilov & Procaccia Reference L'vov, Podivilov and Procaccia1997). However, the difficulties encountered were severe enough to thwart progress in this direction. We refer to Smith & Woodruff (Reference Smith and Woodruff1998), Adzhemyan, Antonov & Vasil'ev (Reference Adzhemyan, Antonov and Vasil'ev1999) and Zhou (Reference Zhou2010) for reviews of these developments.

In the meantime, a novel formulation of the RG has emerged, which allows for non-perturbative approximation schemes, and thereby bypasses the need for a small parameter. The FRG is a modern implementation of Wilson's original conception of the RG (Wilson & Kogut Reference Wilson and Kogut1974). It was formulated in the early nineties (Wetterich Reference Wetterich1993; Ellwanger Reference Ellwanger1994; Morris Reference Morris1994), and widely developed since then (Berges, Tetradis & Wetterich Reference Berges, Tetradis and Wetterich2002; Kopietz, Bartosch & Schütz Reference Kopietz, Bartosch and Schütz2010; Delamotte Reference Delamotte2012; Dupuis et al. Reference Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier and Wschebor2021). One of the noticeable features of this formalism is its versatility, as testified by its wide range of applications, from high-energy physics (quantum chromodynamics and quantum gravity) to condensed matter (fermionic and bosonic quantum systems) and classical statistical physics, including non-equilibrium classical and quantum systems or disordered ones. We refer the interested reader to Dupuis et al. (Reference Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier and Wschebor2021) for a recent review. This has led to fertile methodological transfers, borrowing from one area to the other. The FRG was moreover promoted to a high-precision method, since it was shown to yield, for the archetypical three-dimensional Ising model, results for the critical exponents competing with the best available estimates in the literature (Balog et al. Reference Balog, Chaté, Delamotte, Marohnić and Wschebor2019), and to yield, for the ${{O}}(N)$ models in general, the most precise estimates for the critical exponents (De Polsi et al. Reference De Polsi, Balog, Tissier and Wschebor2020; De Polsi, Hernández-Chifflet & Wschebor Reference De Polsi, Hernández-Chifflet and Wschebor2021).

The FRG has been applied to study turbulence in several works (Tomassini Reference Tomassini1997; Mejía-Monasterio & Muratore-Ginanneschi Reference Mejía-Monasterio and Muratore-Ginanneschi2012; Barbi & Münster Reference Barbi and Münster2013; Canet, Delamotte & Wschebor Reference Canet, Delamotte and Wschebor2016; Canet et al. Reference Canet, Rossetto, Wschebor and Balarac2017; Tarpin, Canet & Wschebor Reference Tarpin, Canet and Wschebor2018; Tarpin et al. Reference Tarpin, Canet, Pagani and Wschebor2019; Pagani & Canet Reference Pagani and Canet2021), including a study of decaying turbulence within a perturbative implementation of the FRG (Fedorenko, Doussal & Wiese Reference Fedorenko, Doussal and Wiese2013). This method has turned out to be fruitful in this context, which is the motivation for this Perspectives article.

2.2. Hydrodynamical equations

The starting point of field-theoretical methods is a ‘microscopic model.’ For fluids, this model is the fundamental hydrodynamical description provided by the NS equation

(2.1)\begin{equation} \partial_t {{v}}_\alpha+ {{v}}_\beta \partial_\beta {{v}}_\alpha={-}\frac 1\rho \partial_\alpha {\rm \pi}+\nu \nabla^{2} {{v}}_\alpha+f_\alpha, \end{equation}

where the velocity field $\boldsymbol {v}$, the pressure field ${\rm \pi}$ and the external force $\boldsymbol {f}$ depend on the space–time coordinates $(t,\boldsymbol {x})$, and with $\nu$ the kinematic viscosity and $\rho$ the density of the fluid. We focus in this review on incompressible flows, satisfying

(2.2)\begin{equation} \partial_\alpha {{v}}_\alpha = 0. \end{equation}

The external stirring force is introduced to maintain a stationary turbulent state. Since the small-scale (i.e. $\ell \ll L$, that is inertial and dissipative) properties are expected to be universal with respect to the large-scale forcing, it can be chosen as a stochastic force, with a Gaussian distribution, of zero average and covariance

(2.3)\begin{equation} \langle f_\alpha(t,\boldsymbol{x})f_\beta(t',\boldsymbol{x}')\rangle=2 \delta(t-t'){N}_{\alpha \beta}\left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right), \end{equation}

where $\boldsymbol{\mathsf{N}}$ is concentrated around the integral scale $L$, such that it models the most common physical situation, where the energy is injected at large scales. This is one of the great advantages of the FRG approach compared with perturbative RG: it can incorporate any functional form for the forcing, and not necessarily a power law. Hence, one can consider a large-scale forcing, and therefore completely bypass the difficulties encountered in perturbative approaches which arise from trying to access the physical situation as an ill-defined limit from a power-law forcing.

We are also interested in a passive scalar field $\theta (t,\boldsymbol {x})$ advected by a turbulent flow. The dynamics of the scalar is governed by the advection–diffusion equation

(2.4)\begin{equation} \partial_{t}\theta +{{v}}_\beta \partial_{\beta}\theta - \kappa_\theta \nabla^{2}\theta = f_\theta, \end{equation}

where $\kappa _\theta$ is the molecular diffusivity of the scalar, and $f_\theta$ is an external stirring force acting on the scalar, which can also be chosen to be Gaussian distributed with zero mean and covariance

(2.5)\begin{equation} \langle f_\theta(t,\boldsymbol{x})f_\theta(t',\boldsymbol{x}')\rangle=2 \delta(t-t')M\left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L_\theta}\right), \end{equation}

with $L_\theta$ the integral scale of the scalar.

Finally, we consider a simplified model of turbulence, introduced by Burgers (Reference Burgers1948), which describes the dynamics of a 1-D compressible randomly stirred fluid. The Burgers equation reads

(2.6)\begin{equation} \partial_t {{v}} + {{v}} \partial_x {{v}} = \nu \partial_x^{2} {{v}} + f , \end{equation}

and can be interpreted as a model for fully compressible hydrodynamics, or the pressureless NS equation (see Bec & Khanin Reference Bec and Khanin2007 for a review). Here, $f$ is again a random Gaussian force with covariance

(2.7)\begin{equation} \langle f(t,x)f(t',x')\rangle=2 \delta(t-t')D(x-x'). \end{equation}

It corresponds to model C of Forster et al. (Reference Forster, Nelson and Stephen1977). In fact, there exists an exact mapping between the Burgers equation and the Kardar–Parisi–Zhang (KPZ) equation which describes the kinetic roughening of a stochastically growing interface (Kardar, Parisi & Zhang Reference Kardar, Parisi and Zhang1986). The KPZ equation gives the time evolution of the height field $h(t,\boldsymbol {x})$ of a $(d-1)$-dimensional interface growing in a $d$-dimensional space as

(2.8)\begin{equation} \partial_t h = \nu \boldsymbol{\nabla}^{2} h + \dfrac \lambda 2 \left(\boldsymbol{\nabla} h\right)^{2} + \eta , \end{equation}

and has become a fundamental model in statistical physics for non-equilibrium scaling phenomena and phase transitions, akin to the Ising model at equilibrium (Halpin-Healy & Zhang Reference Halpin-Healy and Zhang1995; Krug Reference Krug1997; Takeuchi Reference Takeuchi2018). For the standard KPZ equation, $\eta$ is interpreted as a microscopic (small-scale) noise which is delta correlated also in space

(2.9)\begin{equation} \langle \eta(t,\boldsymbol{x})\eta(t',\boldsymbol{x}')\rangle=2 D\delta(t-t')\delta^{d}(\boldsymbol{x}-\boldsymbol{x}'), \end{equation}

but it can be generalised to include a long-range noise $D(|\boldsymbol {x}-\boldsymbol {x}'|)$. Defining the velocity field $\boldsymbol {v} = -\lambda \boldsymbol {\nabla } h$, one obtains from (2.8) the $d$-dimensional generalisation of the Burgers equation (2.6) for a potential flow, with forcing $\boldsymbol {f}=-\lambda \boldsymbol {\nabla } \eta$.

Equations (2.1), (2.4) and (2.6) all yield for some parameters a turbulent regime, where the velocity, pressure or scalar fields undergo rapid and random variations in space and time. One must account for these fluctuations to build a statistical theory of turbulence. A natural way to achieve this is via a path integral, which includes all possible trajectories weighted by their respective probability. How to write such a path integral is explained in § 3.1.

3. Field-theoretical formalism for turbulence

3.1. Path integral representation of a stochastic dynamical equation

The random forcing in the stochastic partial differential equations (2.1), (2.4) and (2.6) acts as a noise source, and thus these stochastic equations are formally equivalent to a Langevin equation. The fundamental difference is that, in usual Langevin description, the origin of the noise lies at the microscopic scale, it is introduced to model some microscopic collision processes, and one is usually interested in the statistical properties of the system at large scales. In the stochastic NS equation, the randomness is introduced at the integral scale, and one is interested in the statistical properties of the system at small scales (but large with respect to the microscopic scale $\varLambda$).

Despite this conceptual difference, in both cases, the dynamical fields are fluctuating ones, and there exists a well-known procedure to encompass all the stochastic trajectories within a path integral, which is the Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD) formalism, (Martin et al. Reference Martin, Siggia and Rose1973; de Dominicis Reference de Dominicis1976; Janssen Reference Janssen1976). The idea is simple, and since it is the starting point of all the subsequent analysis, it is useful to describe the procedure in the simplest case of a scalar field $\varphi (t,\boldsymbol {x})$, following the generic Langevin equation

(3.1)\begin{equation} \partial_t \varphi(t,\boldsymbol{x}) ={-}{\mathcal{F}}[\varphi(t,\boldsymbol{x})] + \eta (t,\boldsymbol{x}) , \end{equation}

where ${\mathcal {F}}$ represents the deterministic forces and $\eta$ the stochastic noise. ${\mathcal {F}}$ is a functional of the fields and their spatial derivatives. The noise has a Gaussian distribution with zero average and a correlator of the form

(3.2)\begin{equation} \langle \eta(t,\boldsymbol{x})\eta(t',\boldsymbol{x}')\rangle= 2 \delta(t-t') D(|\boldsymbol{x}-\boldsymbol{x}'|). \end{equation}

The probablity distribution of the noise is thus given by

(3.3)\begin{equation} {\mathcal{P}}[\eta] = {\mathcal{N}} \exp\left(-\dfrac{1}{4}\int_{t,\boldsymbol{x},\boldsymbol{x}'} \eta(t,\boldsymbol{x})D^{{-}1}(|\boldsymbol{x}-\boldsymbol{x}'|)\eta(t,\boldsymbol{x}') \right) , \end{equation}

with ${\mathcal {N}}$ a normalisation constant. Note that the derivation we present can be generalised to include temporal correlations, or a field dependence into the noise correlations (3.2). The path integral representation of the stochastic equation is obtained in the following way. The probability distribution of the trajectories of the field follows from an average over the noise as

(3.4)\begin{equation} {\mathcal{P}}[\varphi] = \int {\mathcal{D}} \eta {\mathcal{P}}[\eta] \delta(\varphi-\varphi_{\eta}) , \end{equation}

where $\varphi _\eta$ is a solution of (3.1) for a given realisation of $\eta$. A change of variable allows one to replace the constraint $\varphi -\varphi _\eta =0$ by the explicit equation of motion

(3.5)\begin{equation} {\mathcal{G}}[\varphi(t,\boldsymbol{x})] = 0 = \partial_t \varphi(t,\boldsymbol{x}) +{\mathcal{F}}[\varphi(t,\boldsymbol{x})] - \eta (t,\boldsymbol{x}) , \end{equation}

which introduces the functional Jacobian ${\mathcal {J}}[\varphi ] = |{\delta {\mathcal {G}}}/{\delta \varphi }|$. (Two remarks are in order here. First, the existence and uniqueness of the solution of (3.1) has been implicitly assumed. Actually, only a solution in a weak sense is required. For the NS equation, even the existence and uniqueness of weak solutions is a subtle issue from a mathematical viewpoint, and uniqueness may not hold in some cases (Buckmaster & Vicol Reference Buckmaster and Vicol2019). However, uniqueness is not strictly required in this derivation, in the sense that for a typical set of initial conditions, there may exist a set of non-unique velocity configurations, provided they are of zero measure. Second, the expression of the Jacobian ${\mathcal {J}}[\varphi ]$ depends on the discretisation of the Langevin equation (3.1). In Ito's scheme, ${\mathcal {J}}$ is independent of the fields and can be absorbed in the normalisation of ${\mathcal {P}}[\varphi ]$, while in Stratonovich's convention, it depends on the fields, and it can be expressed introducing two Grassmann anti-commuting fields $\psi$ and $\bar \psi$ as

(3.6)\begin{equation} {\mathcal{J}}[\varphi]= \left|{\rm det}\frac{\delta{\mathcal{G}}(t,\boldsymbol{x})}{\delta\varphi(t',\boldsymbol{x}')}\right|= \int {\mathcal{D}}\psi {\mathcal{D}}\bar\psi \exp\left({\int_{t,t',\boldsymbol{x},\boldsymbol{x}'} \bar\psi \frac{\delta {\mathcal{G}}}{\delta \varphi} \psi}\right). \end{equation}

This representation just follows from Gaussian integration of Grassmann variables, which yields the determinant of the operator in the quadratic form (here ${\delta {\mathcal {G}}}/{\delta \varphi }$) rather than its inverse, as for standard (non-Grassmann) variables (Zinn-Justin Reference Zinn-Justin2002). For an additive noise, which means that the noise part in (3.1) and its covariance (3.2) do not depend on the field $\varphi$, the statistical properties of the system are not sensitive to the choice of the discretisation scheme, and both the Ito and Stratonovich conventions yield the same results. We shall here mostly use Ito's discretisation for convenience and omit the Jacobian contribution henceforth.) This leads to

(3.7)\begin{equation} {\mathcal{P}}[\varphi] = \int {\mathcal{D}} \eta {\mathcal{P}}[\eta]{\mathcal{J}}[\varphi] \delta( {\mathcal{G}}[\varphi]) . \end{equation}

One can then use the Fourier representation of the functional Dirac deltas in (3.7), e.g. $\delta (\psi ) = \int {\mathcal {D}}\psi \exp ({-{\rm i} \int \bar \varphi \psi })$, where the conjugate Fourier variable is now a field, denoted with an overbar, and is called the auxiliary field, or response field. Thus, introducing the auxiliary field $\bar \varphi$ in (3.7) yields

(3.8)\begin{equation} {\mathcal{P}}[\varphi] = \int {\mathcal{D}} \eta {\mathcal{P}}[\eta] {\mathcal{D}}\bar\varphi \exp\left({-{\rm i}\int_{t,\boldsymbol{x}} \bar \varphi {\mathcal{G}}[\varphi]}\right) = \int {\mathcal{D}} \bar\varphi \exp({-{\mathcal{S}}[\varphi,\bar\varphi]}). \end{equation}

The second equality stems from the integration over the Gaussian noise $\eta$, resulting in the action

(3.9)\begin{equation} {\mathcal{S}}[\varphi,\bar\varphi] = {\rm i} \int_{t,\boldsymbol{x}} \left\{ \bar\varphi \left(\partial_t\varphi +{\mathcal{F}}[\varphi] \right) \right\} + \int_{t,\boldsymbol{x},\boldsymbol{x}'} \bar \varphi(t,\boldsymbol{x}) D(|\boldsymbol{x}-\boldsymbol{x}'|) \bar\varphi(t,\boldsymbol{x}') . \end{equation}

One usually absorbs the complex $i$ into a redefinition of the auxiliary field $\bar \varphi \to -{\rm i}\bar \varphi$. The action resulting from the MSRJD procedure exhibits a simple structure. The response field appears linearly as a Lagrange multiplier for the equation of motion, while the characteristics of the noise, namely its correlator, are encoded in the quadratic term in $\bar \varphi$.

The path integral formulation offers a simple way to compute all the correlation and response functions of the model. Their generating functional is defined by

(3.10)$$\begin{gather} {\mathcal{Z}}[J,\bar{J}] = \left\langle \exp\left({\int_{t,\boldsymbol{x}} \left\{ J\varphi+\bar{J}\overline{\varphi} \right\}}\right)\right\rangle = \int {\mathcal{D}}\varphi {\mathcal{P}}[\varphi] \exp\left({\int_{t,\boldsymbol{x}} \left\{ J\varphi+\bar{J}\overline{\varphi}\right\}}\right) \nonumber\\ = \int{\mathcal{D}}\bar\varphi {\mathcal{D}}\varphi \exp\left({-{\mathcal{S}}[\varphi,\bar\varphi]+ \int_{t,\boldsymbol{x}} \left\{ J\varphi+\bar{J}\overline{\varphi}\right\}}\right), \end{gather}$$

where $J,\bar {J}$ are the sources for the fields $\varphi,\bar \varphi$ respectively. Correlation functions are obtained by taking functional derivatives of ${\mathcal {Z}}$ with respect to the corresponding sources, e.g. 

(3.11)\begin{equation} \left\langle \varphi(t,\boldsymbol{x}) \right\rangle= \frac{\delta {\mathcal{Z}}}{\delta J(t,\boldsymbol{x})} , \end{equation}

and functional derivatives with respect to response sources generate response functions (i.e. response to an external drive), hence the name ‘response fields.’ We shall henceforth use ‘correlations’ in a generalised sense including both correlation and response functions. In equilibrium statistical mechanics, ${\mathcal {Z}}$ embodies the partition function of the system, while in probability theory, it is called the characteristic function, which is the generating function of moments.

One usually also considers the functional ${\mathcal {W}} = \ln {\mathcal {Z}}$, which is the analogue of a Helmholtz free energy in the context of equilibrium statistical mechanics. In probability theory, it is the generating function of cumulants. When fields rather than variables are involved, the cumulants are generalised to connected correlation functions, and hence ${\mathcal {W}}$ is the generating functional of connected correlation functions, for instance,

(3.12)\begin{equation} \left\langle \varphi(t,\boldsymbol{x})\varphi(t',\boldsymbol{x}')\right\rangle_c \equiv \left\langle \varphi(t,\boldsymbol{x})\varphi(t',\boldsymbol{x}')\right\rangle - \left\langle \varphi(t,\boldsymbol{x})\right\rangle \left\langle \varphi(t',\boldsymbol{x}')\right\rangle = \frac{\delta {\mathcal{W}}}{\delta J(t,\boldsymbol{x})\delta J(t',\boldsymbol{x}')} . \end{equation}

More generally, one can obtain a multi-point (denoted n-point in the following, where n refers to the number of fields involved) generalised connected correlation function ${\mathcal {W}}^{(n)}$ by taking $n$ functional derivatives with respect to either $J$ or $\bar J$ evaluated at $n$ different space–time points $(t_i,\boldsymbol {x}_i)$ as

(3.13)\begin{equation} {\mathcal{W}}^{(n)}(t_1,\boldsymbol{x}_1,\ldots,t_n,\boldsymbol{x}_n) = \frac{\delta^{n} {\mathcal{W}}}{\delta {\mathcal{J}}_{i_1}(t_1,\boldsymbol{x}_1)\cdots \delta {\mathcal{J}}_{i_{n}}(t_{n},\boldsymbol{x}_n)} , \end{equation}

where ${\mathcal {J}} = (J,\bar J)$ and hence $i_k\in \{1,2\}$. It is also useful to introduce another notation ${\mathcal {W}}^{(\ell,m)}$, which specifies that the $\ell$ first derivatives are with respect to $J$ and the $m$ last to $\bar J$, that is

(3.14)\begin{align} {\mathcal{W}}^{(\ell,m)}(t_1,\boldsymbol{x}_1,\ldots,t_{\ell+m},\boldsymbol{x}_{\ell+m}) = \frac{\delta^{\ell+m} {\mathcal{W}}}{\delta J(t_1,\boldsymbol{x}_1)\cdots \delta J(t_\ell,\boldsymbol{x}_\ell)\delta \bar J(t_{\ell+1},\boldsymbol{x}_{\ell+1})\cdots \delta \bar J(t_{\ell+m},\boldsymbol{x}_{\ell+m})}. \end{align}

A last generating functional which plays a central role in field theory and in the FRG framework is the Legendre transform of ${\mathcal {W}}$, i.e. the analogue of the Gibbs free energy in equilibrium statistical mechanics, defined as

(3.15)\begin{equation} \varGamma[\langle\varphi\rangle,\langle\bar\varphi\rangle] = {\sup}_{\left\{J,\bar J\right\}} \left[\int_{t,\boldsymbol{x}} \left\{ J \langle \varphi\rangle + \bar{J} \langle \bar\varphi\rangle \right\} -{\mathcal{W}}[J,\bar{J}] \right]. \end{equation}

In field theory, $\varGamma$ is called the effective action, it is a functional of the average fields, defined with the usual Legendre conjugate relations

(3.16a,b)\begin{equation} \langle \varphi(t,\boldsymbol{x})\rangle = \frac{\delta {\mathcal{W}}}{\delta J(t,\boldsymbol{x})},\quad J(t,\boldsymbol{x})\ = \frac{\delta \varGamma}{\delta \langle \varphi(t,\boldsymbol{x})\rangle}, \end{equation}

and similarly for the response fields. From a field-theoretical viewpoint, $\varGamma$ is the generating functional of one-particle-irreducible correlation functions, which are obtained by taking functional derivatives of $\varGamma$ with respect to average fields

(3.17)\begin{equation} {\varGamma}^{(n)}(t_1,\boldsymbol{x}_1,\ldots,t_n,\boldsymbol{x}_n) = \frac{\delta^{n} {\varGamma}}{\delta {\varPsi}_{i_1}(t_1,\boldsymbol{x}_1)\cdots \delta {\varPsi}_{i_{n}}(t_{n},\boldsymbol{x}_n)} , \end{equation}

where ${\varPsi } = (\langle \varphi \rangle,\langle \bar \varphi \rangle )$ and $i_k\in \{1,2\}$. They are also denoted ${\varGamma }^{(\ell,m)}$ conforming to the definition (3.14). The one-particle-irreducible correlation functions are also simply called vertices, because in diagrammatic representations à la Feynman they precisely correspond to the vertices of the diagrams, as for instance in figure 1. An important point to be highlighted is that both sets of correlation functions $\varGamma ^{(n)}$ and ${\mathcal {W}}^{(n)}$ contain the exact same information on the statistical properties of the model. Each set can be simply reconstructed from the other via a sum of tree diagrams.

Figure 1. Diagrammatic representation of the flow of $\bar \varGamma _{\kappa }^{(2)}$ given by (5.5). The bold lines represent the propagator $\bar G_\kappa$, the crosses the derivative of the regulator $\partial _\kappa {\mathcal {R}}_\kappa$ and the hatched dots the vertices $\bar \varGamma ^{(n)}_\kappa$ with $n$ the number of legs. The loop indicates that all the internal indices are summed over along the loop, and the internal momentum $\boldsymbol {q}$ and frequency $\omega$ circulating in the loop are integrated over.

To conclude these general definitions, we also consider the Fourier transforms in space and time of all these correlation functions. The Fourier convention, used throughout this Perspectives article, is

(3.18)\begin{align} f(\omega,\boldsymbol{q}) = \int_{t,\boldsymbol{x}} f({t,\boldsymbol{x}}) \exp\left({-{\rm i} \boldsymbol{q} \boldsymbol{\cdot}\boldsymbol{x} + {\rm i}\omega t}\right),\quad f(t,\boldsymbol{x}) = \int_{\omega,\boldsymbol{q}} f (\omega,\boldsymbol{q}) \exp\left({{\rm i} \boldsymbol{q} \boldsymbol{\cdot}\boldsymbol{x} - {\rm i}\omega t}\right), \end{align}

where $\int _{t,\boldsymbol {x}}\equiv \int \,{\rm d}t \,{\rm d}^{d} \boldsymbol {x}$ and $\int _{\omega,\boldsymbol {q}} \equiv \int ({{\rm d}^{d} \boldsymbol {q}}/{(2{\rm \pi} )^{d}})({{\rm d}\omega }/{2{\rm \pi} })$. Because of translational invariance in space and time, the Fourier transform of an $n$-point correlation function, e.g. $\varGamma ^{(n)}$, takes the form

(3.19)\begin{align} \varGamma^{(n)}({\omega_1,\boldsymbol{p}_1}, \ldots,\omega_n, {\boldsymbol{p}_{n}}) = (2{\rm \pi})^{d+1}\delta\left(\sum_i \omega_i\right)\delta^{d}\left(\sum_i \boldsymbol{p}_i\right) \bar\varGamma^{(n)}({\omega_1,\boldsymbol{p}_1}, \ldots, {\omega_{n-1},\boldsymbol{p}_{n-1}}) , \end{align}

that is, the total wavevector and total frequency are conserved, and the last frequency–wavevector arguments $\omega _n,\boldsymbol {p}_n$ can be omitted since they are fixed as minus the sum of the others.

3.2. Action for the hydrodynamical equations

The MSRJD procedure can be straightforwardly applied to the ($d$-dimensional) Burgers equation where the field $\varphi$ is replaced by the velocity field $\boldsymbol {v}$. Introducing the response field $\bar {\boldsymbol {v}}$, it yields the action

(3.20)\begin{equation} {\mathcal{S}}_{B} = \int_{t,\boldsymbol{x}}\bar {{v}}_\alpha \left[\partial_t {{v}}_\alpha + {{v}}_\beta\partial_\beta {{v}}_\alpha -\nu \nabla^{2} {{v}}_\alpha \right] - \int_{t,\boldsymbol{x},\boldsymbol{x}'} \bar {{v}}_\alpha D_{\alpha\beta}(\boldsymbol{x}-\boldsymbol{x}')\bar{{v}}_\beta . \end{equation}

For the NS equation, one has to also include the incompressibility constraint (2.2). This can be simply achieved by including the factor $\delta (\partial _\beta v_\beta )$ in (3.4). This functional delta can then be exponentiated along with the equation of motion through the introduction of an additional response field $\overline {{\rm \pi} }$. This yields the following path integral representation:

(3.21)\begin{align} {\mathcal{Z}}[\boldsymbol{J},\overline{\boldsymbol{J}},K,\bar{K}] = \int \mathcal{D}\boldsymbol{v}\mathcal{D}\overline{\boldsymbol{v}}\mathcal{D}{\rm \pi} \mathcal{D}\bar {\rm \pi}\, \exp\left({-{\mathcal{S}}_{NS}[\boldsymbol{v},\overline{\boldsymbol{v}},{\rm \pi},\bar {\rm \pi}] +\int_{t,\boldsymbol{x}}\left\{ \boldsymbol{J}\boldsymbol{\cdot} \boldsymbol{v}+\overline{\boldsymbol{J}}\boldsymbol{\cdot} \overline{\boldsymbol{v}}+K {\rm \pi}+\bar K\bar {\rm \pi}\right\}}\right) , \end{align}

for the fluctuating velocity and pressure fields and their associated response fields, with the NS action

(3.22)$$\begin{gather} {\mathcal{S}}_{NS}[\boldsymbol{v},\bar{\boldsymbol{v}},{\rm \pi},\bar {\rm \pi}] = \int_{t,\boldsymbol{x}}\left\{\bar {\rm \pi}\partial_\alpha {{v}}_\alpha + \bar {{v}}_\alpha\left[ \partial_t {{v}}_\alpha -\nu \nabla^{2} {{v}}_\alpha +{{v}}_\beta\partial_\beta {{v}}_\alpha+\frac 1\rho \partial_\alpha {\rm \pi}\right]\right\}\nonumber\\ -\int_{t,\boldsymbol{x},\boldsymbol{x}'}\bar {{v}}_\alpha(t,\boldsymbol{x}) N_{\alpha\beta}\left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right)\bar {{v}}_\beta(t,\boldsymbol{x}') . \end{gather}$$

In this formulation, we have kept the pressure field and introduced a response field $\bar {{\rm \pi} }$ to enforce the incompressibility constraint. Alternatively, the pressure field can be integrated out using the Poisson equation, such that one obtains a path integral in terms of two fields $\boldsymbol {v}$ and $\bar {\boldsymbol {v}}$ instead of four, at the price of a resulting action which is non-local (Barbi & Münster Reference Barbi and Münster2013). We choose here to keep the pressure fields since the whole pressure sector turns out to be very simple to handle, as is shown in Appendix A.

We now consider a passive scalar field $\theta$ transported by a turbulent NS velocity flow according to the advection–diffusion equation (2.4). The associated field theory is simply obtained by adding the two fields $\theta, \bar \theta$ and the two corresponding sources $j,\bar j$ to the field multiplet $\varPhi =(\boldsymbol {v},\bar {\boldsymbol {v}},{\rm \pi},\bar {\rm \pi},\theta,\bar \theta )$ and source multiplet ${\mathcal {J}} = (\boldsymbol {J},\bar {\boldsymbol {J}},K,\bar K,j,\bar j)$ respectively. The generating functional then reads

(3.23)\begin{equation} {\mathcal{Z}}[{\mathcal{J}}] = \int \mathcal{D}\varPhi\exp\left({-{\mathcal{S}}_{NS}[\boldsymbol{v},\bar{\boldsymbol{v}},{\rm \pi},\bar {\rm \pi}] -{\mathcal{S}}_{\theta}[\theta,\bar\theta,\boldsymbol{v}] + \int_{t,\boldsymbol{x}} {\mathcal{J}}_\ell \varPhi_\ell}\right), \end{equation}

with the passive scalar action given by

(3.24)\begin{align} {\mathcal{S}}_{\theta}[\theta,\bar\theta,\boldsymbol{v}] = \int_{t,\boldsymbol{x}}\,\bar{\theta}\left[\partial_{t}\theta+{{v}}_{\beta}\partial_{\beta}\theta-\kappa_\theta \nabla^{2}\theta\right] -\int_{t,\boldsymbol{x} ,\boldsymbol{x}'} \bar{\theta}\left(t,\boldsymbol{x}\right)M\left(\frac{\boldsymbol{x}-\boldsymbol{x}'}{L_\theta}\right)\bar{\theta}\left(t,\boldsymbol{x}'\right). \end{align}

The two actions (3.22) and (3.24) possess fundamental symmetries, some of which have only been recently identified. They can be used to derive a set of exact identities relating among each other different correlation functions of the system. These identities, which include as particular cases the Kármán–Howarth and Yaglom relations, contain fundamental information on the system. We provide in § 3.3 a detailed analysis of these symmetries, and show how the set of exact identities can be inferred from Ward identities.

3.3. Symmetries and extended symmetries

We are interested in the stationary state of fully developed homogeneous and isotropic turbulence. We hence assume translational invariance in space and time, as well as rotational invariance. Beside these symmetries, the NS equation possesses other well-known symmetries, such as the Galilean invariance. In the field-theoretical formulation, these symmetries are obviously carried over, but they are moreover endowed with a systematic framework to be fully exploited. Indeed, the path integral formulation provides the natural tool to express the consequences of the symmetries on the correlation functions of the theory, under the form of exact identities which are called Ward identities.

In fact, the field-theoretical formulation is even more far reaching, in the sense that the very notion of symmetry can be extended. A symmetry means an invariance of the model (equation of motion or action) under a given transformation of the fields and space–time coordinates. Identifying the symmetries of a model is very useful because it allows for simplification, and also uncovering of invariants, as according to Noether's theorem, continuous symmetries are associated with conserved quantities. In general, the symmetries considered are global symmetries, i.e. their parameters are constants (e.g. angle of a global rotation, vector of a global translation). However, if one can identify local symmetries, i.e. whose parameters are space and/or time dependent, this will lead to local conservation laws, which are much richer, much more constraining. In general, the global symmetries of the model cannot be simply promoted to local symmetries by letting their parameters depend on space and/or time. However, this can be achieved in certain cases at the price of extending the notion of symmetry. More precisely, it is useful to also consider transformations that do not leave the model strictly invariant, but ‘almost’, in the sense that they lead to a variation which is at most linear in the fields. We refer to them as extended symmetries. The key is that one can deduce from them exact local identities, which are thus endowed with a stronger content than the original global identities associated with global symmetries, i.e. they provide additional constraints.

One may wonder why one focuses on linear variations specifically and what is special about them. In fact, in principle, one can consider any transformation of the fields and coordinates and just write down the corresponding variation of the action. Once inserted in the path integral (3.10), this leads to a relation for the average value of this variation. The reason is that only when it is linear in the fields can this relation be turned into an identity for the generating functionals themselves. In this case, it becomes extremely powerful because it allows one to obtain an infinite set of exact relations between correlation functions by taking functional derivatives. The mechanism is very simple, and is exemplified in the simplest case of the symmetries of the pressure sector in Appendix A.1. In the following, we detail two specific extended symmetries, since they play a fundamental role in § 7; refer to Appendix A for additional ones. Indeed, these two symmetries yield the identities (3.31) and (3.36), which allow for an exact closure at large wavenumbers of the FRG equations.

3.3.1. Time-dependent Galilean symmetry

A fundamental symmetry of the NS equation is the Galilean invariance, which is the invariance under the global transformation $\boldsymbol {x}\to \boldsymbol {x}'=\boldsymbol {x}+\boldsymbol {v}_0 t$, $\boldsymbol {v}(t,\boldsymbol {x})\to \boldsymbol {v}(t,\boldsymbol {x}')-\boldsymbol {v}_0$. In fact, it was early recognised in the field-theoretical context that a time-dependent, (also called time-gauged) version of this transformation leads to an extended symmetry of the NS action and useful Ward identities (Adzhemyan, Antonov & Kim Reference Adzhemyan, Antonov and Kim1994; Antonov, Borisenok & Girina Reference Antonov, Borisenok and Girina1996; Adzhemyan et al. Reference Adzhemyan, Antonov and Vasil'ev1999). Considering an infinitesimal arbitrary time-dependent vector $\boldsymbol {\varepsilon }(t)$, this transformation reads

(3.25)\begin{equation} \left. \begin{aligned} \delta {{v}}_\alpha(t,\boldsymbol{x}) & ={-}\dot{\varepsilon}_\alpha(t)+\varepsilon_\beta(t) \partial_\beta {{v}}_\alpha(t,\boldsymbol{x})\\ \delta \varPhi_k(t,\boldsymbol{x}) & =\varepsilon_\beta(t) \partial_\beta \varPhi_k(t,\boldsymbol{x}), \end{aligned} \right\} \end{equation}

where $\dot {\varepsilon }_\alpha = {\textrm {d}\varepsilon _\alpha }/{\textrm {d}t}$, and $\varPhi _k$ denotes any other fields $\bar {\boldsymbol {v}},{\rm \pi},\bar {{\rm \pi} },\ldots$. The variation $\varepsilon _\beta (t)\partial _\beta (\,{{\cdot }}\,)$ just originates from the change of coordinates. The global Galilean transformation is recovered for a constant velocity $\dot {\boldsymbol {\varepsilon }}(t) = \boldsymbol {v}_0$.

The overall variation of the NS action under the transformation (3.25) is

(3.26)\begin{equation} \delta {\mathcal{S}}_{NS}={-}\int_{t,\boldsymbol{x}} {\varepsilon}_\alpha(t) \partial_t^{2} \bar {{v}}_\alpha(t,\boldsymbol{x}). \end{equation}

Since the field transformation (3.25) is a mere affine change of variable in the functional integral ${\mathcal {Z}}$, it must leave it unaltered. Performing this change of variable in (3.21) and expanding the exponential to first order in $\varepsilon$, one obtains

(3.27)\begin{equation} \left\langle \delta {\mathcal{S}}_{NS}\right\rangle = \left\langle \delta \int_{t,\boldsymbol{x}}{\mathcal{J}}_\ell \varPhi_\ell\right\rangle = \int_{t,\boldsymbol{x}} \left\{ -\dot\varepsilon_\alpha(t) J_\alpha(t,\boldsymbol{x}) + \varepsilon_\beta(t) {\mathcal{J}}_\ell(t,\boldsymbol{x}) \partial_\beta \varPsi_\ell(t,\boldsymbol{x}) \right\} . \end{equation}

Since this identity is valid for arbitrary $\boldsymbol {\varepsilon }(t)$, one deduces

(3.28)\begin{equation} \int_{\boldsymbol{x}} \left\{\partial_t J_\alpha(t,\boldsymbol{x}) + {\mathcal{J}}_\ell(t,\boldsymbol{x}) \partial_\alpha \varPsi_\ell(t,\boldsymbol{x}) \right\} ={-}\int_{\boldsymbol{x}} \partial_t^{2} \bar u_\alpha(t,\boldsymbol{x}) . \end{equation}

We use throughout this paper the notation $\boldsymbol {u} \equiv \langle \boldsymbol {v}\rangle$ and $\bar {\boldsymbol {u}} \equiv \langle \bar {\boldsymbol {v}}\rangle$ for the average fields, and generically for the field multiplet $\varPsi \equiv \langle \varPhi \rangle$. The identity (3.28) is integrated over space, but local in time. In contrast, the identity stemming from the usual Galilean invariance with a constant $\boldsymbol {v}_0$ is integrated over time as well, and the right-hand side is replaced by zero since the action is invariant under global Galilean transformation.

One can express the sources in term of derivatives of $\varGamma$ using (3.16a,b) and rewrite the exact identity (3.28) as the following Ward identity for the functional $\varGamma$:

(3.29)\begin{equation} \int_{\boldsymbol{x}} \left\{ \partial_t \frac{\delta \varGamma}{\delta u_\alpha} + \partial_\alpha \varPsi_\ell \frac{\delta \varGamma}{\delta \varPsi_\ell}\right\} ={-}\int_{\boldsymbol{x}} \partial_t^{2} \bar u_\alpha . \end{equation}

Note that replacing instead the average values of the fields using (3.16a,b), the identity (3.28) can be equivalently written for the functional ${\mathcal {W}}$ as

(3.30)\begin{equation} \int_{\boldsymbol{x}} \left\{\partial_t J_\alpha + {\mathcal{J}}_\ell \partial_\alpha \frac{\delta {\mathcal{W}}}{\delta {\mathcal{J}}_\ell} \right\} ={-}\int_{\boldsymbol{x}} \partial_t^{2} \frac{\delta {\mathcal{W}}}{\delta \bar J_\alpha}. \end{equation}

The identities (3.29) and (3.30) are functional in the fields. One can deduce from them, by functional differentiation, an infinite set of exact identities amongst the correlation functions $\varGamma ^{(n)}$ or ${\mathcal {W}}^{(n)}$. Let us express them for the vertices. They are obtained by taking functional derivatives of the identity (3.29) with respect to $\ell$ velocity and $m$ response velocity fields, and setting the fields to zero, which yields in Fourier space

(3.31)\begin{align} &\varGamma^{(\ell+1,m)}_{\alpha\alpha_1\ldots\alpha_{\ell+m}}(\omega,\boldsymbol{p}=\boldsymbol{0}, \omega_1,\boldsymbol{p}_1, \ldots , \omega_{\ell+m},\boldsymbol{p}_{\ell+m})\nonumber\\ &\qquad={-}\sum_{k=1}^{\ell+m}\frac{p_k^{\alpha}}{\omega}\varGamma^{(\ell,m)}_{\alpha_1\ldots\alpha_{\ell+m}}(\omega_1,\boldsymbol{p}_1,\ldots,\omega_k+\omega,\boldsymbol{p}_k,\ldots, \omega_{\ell+m},\boldsymbol{p}_{\ell+m})\nonumber\\ &\qquad \equiv {\mathcal{D}}_\alpha(\omega) \varGamma^{(\ell,m)}_{\alpha_1\ldots\alpha_{\ell+m}}(\omega_1,\boldsymbol{p}_1, \ldots, \omega_{\ell+m},\boldsymbol{p}_{\ell+m}), \end{align}

where the $\alpha _i$ are the space indices of the vector fields. We refer to Tarpin et al. (Reference Tarpin, Canet and Wschebor2018) for details on the derivation. The operator ${\mathcal {D}}_\alpha (\omega )$ hence successively shifts by $\omega$ all the frequencies of the function on which it acts. The identities (3.31) exactly relate an arbitrary $(\ell +m+1)$-point vertex function with one vanishing wavevector carried by a velocity field $u_\alpha$ to a lowered-by-one order $(\ell +m)$-point vertex function. It is clear that this type of identity can constitute a key asset to address the closure problem of turbulence, since the latter precisely requires us to express higher-order statistical moments in terms of lower-order ones. Of course, this relation only fixes $\varGamma ^{(\ell +m+1)}$ in a specific configuration, namely with one vanishing wavevector, so it does not allow one to completely eliminate this vertex, and it is not obvious a priori how it can be exploited. In fact, we will show that, within the FRG framework, this type of configuration plays a dominant role at small scales, and the exact identities (3.31) (together with (3.35)) in turn yields the closure of the FRG equations in the corresponding limit.

3.3.2. Shift of the response fields

It was also early noticed in the field-theoretical framework that the NS action is invariant under a constant shift of the velocity response fields (as by integration by parts in (3.22) this constant can be eliminated). However, it was not identified until recently that this symmetry could also be promoted to a time-dependent one (Canet, Delamotte & Wschebor Reference Canet, Delamotte and Wschebor2015). The latter corresponds to the following infinitesimal coupled transformation of the response fields:

(3.32)\begin{equation} \left. \begin{aligned} \delta \bar {{v}}_\alpha(t,\boldsymbol{x}) & =\bar \varepsilon_\alpha(t)\\ \delta \bar p(t,\boldsymbol{x}) & = {{v}}_\beta(t,\boldsymbol{x}) \bar \varepsilon_\beta(t) . \end{aligned} \right\} \end{equation}

This transformation indeed induces a variation of the NS action which is only linear in the fields

(3.33)\begin{equation} \delta {\mathcal{S}}_{NS} = \int_{t,\boldsymbol{x}} {\bar\varepsilon}_\beta(t) \partial_t {{v}}_\beta(t,\boldsymbol{x}) +2 \int_{t,\boldsymbol{x},\boldsymbol{x}'} {\bar\varepsilon}_\alpha(t) N_{\alpha_\beta} \left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right) \bar {{v}}_\beta(t,\boldsymbol{x}') . \end{equation}

Hence, interpreted as a change of variable in (3.21), this yields the identity $\langle \delta {\mathcal {S}}_{NS}\rangle = \langle \delta \int _{t,\boldsymbol {x}}{\mathcal {J}}_\ell \varPhi _\ell \rangle$, which can be written as the following Ward identity for the functional $\varGamma$:

(3.34)\begin{align} \int_{\boldsymbol{x}} \left\{\frac{\delta \varGamma}{\delta \bar u_\alpha(t,\boldsymbol{x})} + u_\alpha(t,\boldsymbol{x}) \frac{\delta \varGamma}{\delta \bar p(t,\boldsymbol{x})} \right\}= \int_{\boldsymbol{x}} \partial_t u_\alpha(t,\boldsymbol{x}) + 2 \int_{\boldsymbol{x},\boldsymbol{x}'}N_{\alpha_\beta} \left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right) \bar u_\beta(t,\boldsymbol{x}') . \end{align}

Note that this identity is again local in time. Taking functional derivatives with respect to velocity and response velocity fields and evaluating at zero fields, one can deduce again exact identities for vertex functions (Canet et al. Reference Canet, Delamotte and Wschebor2016). They give the expression of any $\varGamma ^{(\ell,m)}$ with one vanishing wavevector carried by a response velocity, which simply reads in Fourier space

(3.35)\begin{equation} \varGamma_{\alpha_1\ldots\alpha_{\ell+m}}^{(\ell,m)}(\omega_1,\boldsymbol{p}_1, \ldots,\omega_\ell,\boldsymbol{p}_\ell, \omega_{\ell+1},\boldsymbol{p}_{\ell+1}=\boldsymbol{0},\ldots)= 0 , \end{equation}

for all $(\ell,m)$ except for the two lower-order ones which keep their original form given by ${\mathcal {S}}_{{NS},\alpha \beta }^{(1,1)}$ and ${\mathcal {S}}_{{NS},\alpha \beta \gamma }^{(2,1)}$ respectively, i.e.

(3.36) \begin{equation} \left. \begin{aligned} & \varGamma_{\alpha\beta}^{(1,1)}(\omega_1,\boldsymbol{p}_1, \omega_2,\boldsymbol{p}_2 =\boldsymbol{0})={\rm i}\omega_1\delta_{\alpha\beta} \;(2{\rm \pi})^{d+1}\delta(\omega_1+\omega_2)\delta^{d}(\boldsymbol{p}_1),\\ & \varGamma_{\alpha\beta\gamma}^{(2,1)}(\omega_1,\boldsymbol{p}_1,\omega_2,\boldsymbol{p}_2,\omega_3,\boldsymbol{p}_3 =\boldsymbol{0})) ={-}{\rm i} (p_2^{\alpha} \delta_{\beta\gamma} +{\rm i} p_1^{\beta} \delta_{\alpha\gamma})\\ & \qquad \times(2{\rm \pi})^{d+1}\delta(\omega_1+\omega_2+\omega_3)\delta^{d}(\boldsymbol{p}_1+\boldsymbol{p}_2). \end{aligned} \right\} \end{equation}

Let us emphasise that the analysis of extended symmetries can still be completed. First, for 2-D turbulence, additional extended symmetries have recently been unveiled, which are reported in Appendix C. One of them is also realised in 3-D turbulence but has not been exploited yet in the FRG formalism. Second, another important symmetry of the Euler or NS equation is the scaling or dilatation symmetry, which amounts to the transformation $\boldsymbol {v}(t,\boldsymbol {x}) \to b^{h} \boldsymbol {v}(b^{z} t,b \boldsymbol {x})$ (Frisch Reference Frisch1995; Dubrulle Reference Dubrulle2019). One can also derive from this symmetry, possibly extended, functional Ward identities. This route has not been explored either. Both could lead to future fruitful developments.

3.4. Kármán–Howarth and Yaglom relations from symmetries

The path integral formulation conveys an interesting viewpoint on well-known exact identities such as the Kármán–Howarth relation (von Kármán & Howarth Reference von Kármán and Howarth1938) or the equivalent Yaglom relation for passive scalars (Yaglom Reference Yaglom1949). The Kármán–Howarth relation stems from the energy budget equation associated with the NS equation, upon imposing stationarity, homogeneity and isotropy. From this relation, one can derive the exact four-fifths Kolmogorov law for the third-order structure function $S^{(3)}$ (see e.g. Frisch Reference Frisch1995). It turns out that the Kármán–Howarth relation also emerges as the Ward identity associated with a space–time-dependent shift of the response fields (which is a generalisation of the time-dependent shift discussed in § 3.3.2), and can thus be obtained in the path integral formulation as a consequence of symmetries (Canet et al. Reference Canet, Delamotte and Wschebor2015).

To show this, let us consider the NS action (3.22) with an additional source $L_{\alpha \beta }$ in (3.21) coupled to the local quadratic term ${{v}}_\alpha (t,\boldsymbol {x}){{v}}_\beta (t,\boldsymbol {x})$ (which is a composite operator in the field-theoretical language), i.e. the term $\int _{t,\boldsymbol {x}} {{v}}_\alpha L_{\alpha \beta }{{v}}_\beta$ is added to the source terms. This implies that a local, i.e. at coinciding space–time points, quadratic average of the velocities, can then be simply obtained by taking a functional derivative of ${\mathcal {W}}$ with respect to this new source

(3.37)\begin{equation} \left\langle {{v}}_\alpha(t,\boldsymbol{x}) {{v}}_\beta(t,\boldsymbol{x})\right\rangle = \frac{\delta {\mathcal{W}}}{\delta L_{\alpha\beta}(t,\boldsymbol{x})} ={-} \frac{\delta \varGamma}{\delta L_{\alpha\beta}(t,\boldsymbol{x})} , \end{equation}

where the last equality stems from the Legendre transform relation (3.15). (Note that to define $\varGamma$, the Legendre transform is not taken with respect to the source $L_{\alpha \beta }$, i.e. both functional ${\mathcal {W}}$ and $\varGamma$ depend on $L_{\alpha \beta }$, hence the relation (3.37).)

The introduction of the source for the composite operator ${{v}}_\alpha {{v}}_\beta$ allows one to consider a further extended symmetry of the NS action, namely the time- and space-dependent version of the field transformation (3.32), which amounts to $\bar \varepsilon _\alpha (t)\to \bar \varepsilon _\alpha (t,\boldsymbol {x})$. The variation of the NS action under this transformation writes

(3.38)\begin{align} \langle \delta {\mathcal{S}}_{NS} \rangle=\left\langle \partial_t {{v}}_\alpha+\frac{1}{\rho}\partial_\alpha {\rm \pi}-\nu \nabla^{2} {{v}}_\alpha +\partial_\beta({{v}}_\alpha {{v}}_\beta) -2\int_{\boldsymbol{x}'}\left(N_{\alpha\beta}\left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right) \bar {{v}}_\beta(t,\boldsymbol{x}')\right) \right\rangle, \end{align}

which is linear in the fields, but for the local quadratic term. However, this term can be expressed as a derivative with respect to $L_{\alpha \beta }$ using (3.37). The resulting Ward identity can be written equivalently in terms of $\varGamma$ or ${\mathcal {W}}$. We here write it for ${\mathcal {W}}$ since it renders more direct the connection with the Kármán–Howarth relation

(3.39)$$\begin{gather} -\partial_t \frac{\delta \mathcal{W}}{\delta J_\alpha}-\frac{1}{\rho}\partial_\alpha\frac{\delta \mathcal{W}}{\delta K} +\nu \nabla^{2} \frac{\delta \mathcal{W}}{\delta J_\alpha}+ \bar J_\alpha + \bar K \frac{\delta \mathcal{W}}{\delta J_\alpha} -\partial_\beta \frac{\delta \mathcal{W}}{\delta L_{\alpha\beta}}\nonumber\\ + 2\int_{\boldsymbol{x}'} N_{\alpha\beta}\left(\frac{|\boldsymbol{x}-\boldsymbol{x}'|}{L}\right)\frac{\delta \mathcal{W}}{\delta \bar J_\beta(t,\boldsymbol{x}')} =0 . \end{gather}$$

We emphasise that, compared with (3.34), this identity is now fully local, in space as well as in time, i.e. it is no longer integrated over space. It is also an exact identity for the generating functional $\mathcal {W}$ itself, which means that it entails an infinite set of exact identities amongst correlation functions. The Kármán–Howarth relation embodies the lowest-order one, which is obtained by differentiating (3.39) with respect to $J_\gamma (t_y,\boldsymbol {y})$, and evaluating the resulting identity at zero external sources. Note that in the MSRJD formalism, the term proportional to $N_{\alpha \beta }$ is simply equal to a force–velocity correlation $\langle f_\alpha (t,\boldsymbol {x}) {{v}}_\gamma (t_y,\boldsymbol {y}) \rangle$ (Canet et al. Reference Canet, Delamotte and Wschebor2015). Summing over $\gamma =\alpha$ and specialising to equal time $t_y=t$, one deduces

(3.40)\begin{align} &-\partial_t \langle {{v}}_\alpha(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{y})\rangle + \nu (\Delta_x+\Delta_y) \langle {{v}}_\alpha(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{y})\rangle \nonumber\\ &\qquad -\partial_\beta^{x} \langle {{v}}_\alpha(t,\boldsymbol{x}) {{v}}_\beta(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{y})\rangle -\partial_\beta^{y} \langle {{v}}_\alpha(t,\boldsymbol{y}) {{v}}_\beta(t,\boldsymbol{y}) {{v}}_\alpha(t,\boldsymbol{x})\rangle\nonumber\\ &\qquad+ \langle f_\alpha(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{y}) \rangle+ \langle f_\alpha(t,\boldsymbol{y}) {{v}}_\alpha(t,\boldsymbol{x}) \rangle=0, \end{align}

which is the Kármán–Howarth relation. This fundamental relation is usually expressed in terms of the longitudinal velocity increments $\delta {{v}}_\parallel (\boldsymbol {\ell }) = (\boldsymbol {v}(\boldsymbol {r}+\boldsymbol {\ell }) - \boldsymbol {v}(\boldsymbol{r}))\boldsymbol {{\cdot}} \boldsymbol {\ell}/\ell$ by choosing the two space points as $\boldsymbol {y}=\boldsymbol {x}+\boldsymbol {\ell }$ and $\boldsymbol {x} = \boldsymbol {r}$. Using homogeneity and isotropy, (3.40) can be equivalently expressed as

(3.41)\begin{align} \epsilon(\boldsymbol{\ell}) &\equiv{-}\dfrac 1 4 \nabla_{\boldsymbol{\ell}}\boldsymbol{\cdot} \left\langle |\delta \boldsymbol{v}(\boldsymbol{\ell})|^{2} \delta \boldsymbol{v}(\boldsymbol{\ell}) \right\rangle\nonumber\\ &={-}\dfrac 1 2 \partial_t \left\langle \boldsymbol{v}(\boldsymbol{r}) \boldsymbol{v}(\boldsymbol{r}+\boldsymbol{\ell}) \right\rangle + \left\langle \boldsymbol{v}(\boldsymbol{r})\boldsymbol{\cdot} \dfrac{\boldsymbol{f}(\boldsymbol{r}+\boldsymbol{\ell}) +\boldsymbol{f}(\boldsymbol{r}-\boldsymbol{\ell})}{2} \right\rangle +\nu \nabla_{\ell}^{2} \left\langle \boldsymbol{v}(\boldsymbol{r})\boldsymbol{\cdot} \boldsymbol{v}(\boldsymbol{r}+\boldsymbol{\ell}) \right\rangle. \end{align}

Once again, taking other functional derivatives with respect to arbitrary sources yields infinitely many exact relations. To give another example, by differentiating twice (3.39) with respect to $L_{\mu \nu }(t_y,\boldsymbol {y})$ and $J_\gamma (t_z,\boldsymbol {z})$, one obtains the exact relation for a pressure–velocity correlation (Canet et al. Reference Canet, Delamotte and Wschebor2015)

(3.42)\begin{align} & \nu \langle {{v}}_\alpha(t,\boldsymbol{x}) \Delta_x {{v}}_\alpha(t,\boldsymbol{x}) \boldsymbol{v}^{2}(t,\boldsymbol{y}) \rangle -\frac 1 \rho \partial_\alpha^{x} \langle \boldsymbol{v}^{2}(t,\boldsymbol{y}) {{v}}_\alpha(t,\boldsymbol{x}) {\rm \pi}(t,\boldsymbol{x})\rangle \nonumber\\ &\qquad+ \langle f_\alpha(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{x}) \boldsymbol{v}^{2}(t,\boldsymbol{y}) \rangle - \frac 1 2 \partial^{x}_\alpha \langle {{v}}_\alpha(t,\boldsymbol{x}) \boldsymbol{v}^{2}(t,\boldsymbol{x}) v^{2}(t,\boldsymbol{y}) \rangle =0 , \end{align}

which was first derived in Falkovich, Fouxon & Oz (Reference Falkovich, Fouxon and Oz2010). Taking additional functional derivatives with respect to $\boldsymbol {J}$ or $\boldsymbol{\mathsf{L}}$ (or of the any other sources) generates new exact relations between higher-order correlation functions, involving in the averages one more ${{v}}$ or $\boldsymbol {v}^{2}$ (or any of the other fields) respectively with each derivative compared with (3.40). Although exact, relations for high-order correlation functions may not be of direct practical use since they are increasingly difficult to measure, but they are important at the theoretical level.

Shortly after Kolmogorov's derivation of the exact relation for the third-order structure function, Yaglom established the analogous formula for scalar turbulence (Yaglom Reference Yaglom1949). It was shown in Pagani & Canet (Reference Pagani and Canet2021) that this relation can also be simply inferred from symmetries of the passive scalar action (3.24). More precisely, it ensues from the space–time-dependent shift of the response fields

(3.43a,b)\begin{equation} \bar{\theta}\left(t,\boldsymbol{x}\right)\rightarrow\bar{\theta}\left(t,\boldsymbol{x}\right)+\bar\varepsilon\left(t,\boldsymbol{x}\right),\quad \bar{\rm \pi}\left(t,\boldsymbol{x}\right)\rightarrow \bar{\rm \pi}\left(t,\boldsymbol{x}\right)+\bar \varepsilon\left(t,\boldsymbol{x}\right)\theta\left(t,\boldsymbol{x}\right) \end{equation}

in the path integral (3.23), in the presence of the additional source term $\int _{t,\boldsymbol {x}}L_{\alpha }{{v}}_{\alpha }\theta$, where $L_\alpha$ is the source coupled to the composite operator ${{v}}_{\alpha }\theta$. Following the same reasoning, one deduces an exact functional Ward identity for the passive scalar, which writes in term of ${\mathcal {W}}$

(3.44)\begin{equation} \left(\partial_{t}-{\kappa_\theta}\nabla^{2} \right)\frac{\delta W}{\delta j\left(t,\boldsymbol{x}\right)}+ \partial_{\beta}\frac{\delta W}{\delta L_{\beta}\left(t,\boldsymbol{x}\right)} -\int_{\boldsymbol{y}} M\left(\frac{|\boldsymbol{x}-\boldsymbol{y}|}{L_\theta}\right)\frac{\delta W}{\delta \bar{j}\left(t,\boldsymbol{y}\right)} = 0. \end{equation}

The lowest-order relation stemming from it is the Yaglom relation

(3.45)\begin{equation} -\frac{1}{2}\frac{\partial}{\partial\left(x-y\right)_{\alpha}}\langle\left|\theta\left(t,\boldsymbol{x}\right)-\theta\left(t,\boldsymbol{y}\right)\right|^{2}\left({{v}}_{\alpha}\left(t,\boldsymbol{x}\right)-{{v}}_{\alpha}\left(t,\boldsymbol{y}\right)\right)\rangle = 2\epsilon_{\theta}, \end{equation}

where $\epsilon _\theta$ is the mean dissipation rate of the scalar. As for the Kármán–Howarth relation, it can be re-expressed using homogeneity and isotropy in the more usual form

(3.46)\begin{equation} \langle\left|\theta\left(t,\boldsymbol{r}\right)-\theta\left(t,\boldsymbol{r}+\boldsymbol{\ell}\right)\right|^{2} \delta {{v}}_\parallel(t,\boldsymbol{\ell})\rangle ={-}\dfrac{4}{3}\epsilon_{\theta}\ell. \end{equation}

This relation hence also follows from symmetries in the path integral formulation. Again, an infinite set of higher-order exact relations between scalar and velocity correlation functions can be obtained by further differentiating (3.44) with respect to sources $\boldsymbol {L}$ or $j$.

4. The functional renormalisation group

As mentioned in § 2, the idea underlying the FRG is Wilson's original idea of progressive averaging of fluctuations in order to build up the effective description of a system from its microscopic model. This progressive averaging is organised scale by scale, in general in wavenumber space, and thus leads to a sequence of scale-dependent models, embodied in Wilson's formulation in a scale-dependent Hamiltonian or action (Wilson & Kogut Reference Wilson and Kogut1974). In the FRG formalism, one rather considers a scale-dependent effective action $\varGamma _\kappa$, called the effective average action, where $\kappa$ denotes the RG scale. It is a wavenumber scale, which runs from the microscopic ultraviolet (UV) scale $\varLambda$ to a macroscopic IR scale (e.g. the inverse integral scale $L^{-1}$). The interest of the RG procedure is that the information about the properties of the system can be captured by the flow of these scale-dependent models, i.e. their evolution with the RG scale, without requiring us to explicitly carry out the integration of the fluctuations in the path integral. The RG flow is governed by an exact very general equation, which can take different forms depending on the precise RG used (Wilson & Kogut Reference Wilson and Kogut1974) or Polchinski flow equation (Polchinski Reference Polchinski1984), Callan–Symanzik flow equation (Callan Reference Callan1970; Symanzik Reference Symanzik1970, $\ldots$). The one at the basis of the FRG formalism is usually called the Wetterich equation (Wetterich Reference Wetterich1993). We briefly introduce it in the next sections on the example of the generic scalar field theory of § 3.1, denoting generically $\varPhi (t,\boldsymbol {x})$ the field multiplet, e.g. $\varPhi =(\varphi,\bar \varphi )$, $\varPsi =\langle \varPhi \rangle$ the multiplet of average fields, and ${\mathcal {J}}$ the multiplet of corresponding sources.

4.1. Progressive integration of fluctuations

The core of the RG procedure is to turn the global integration over fluctuations in the path integral (3.10) into a progressive integration, organised by wavenumber shells. To achieve this, one introduces in the path integral a scale-dependent weight $\textrm {e}^{-\Delta {\mathcal {S}}_\kappa }$ whose role is to suppress fluctuations below the RG scale $\kappa$, giving rise to a new, scale-dependent generating functional

(4.1)\begin{equation} {\mathcal{Z}}_\kappa[{\mathcal{J}}] = \int{\mathcal{D}}\varPhi \exp\left({-{\mathcal{S}}[\varPhi] - \Delta{\mathcal{S}}_\kappa[\varPhi]+ \int_{t,\boldsymbol{x}} {\mathcal{J}}_\ell \varPhi_\ell } \right) . \end{equation}

The new term $\Delta {\mathcal {S}}_\kappa$ is chosen quadratic in the fields

(4.2)\begin{equation} \Delta{\mathcal{S}}_\kappa[\varPhi] =\frac 1 2 \int_{t,\boldsymbol{x},\boldsymbol{x}'} \varPhi_m(t,\boldsymbol{x}) {\mathcal{R}}_{\kappa,m m'}(|\boldsymbol{x}-\boldsymbol{x}'|)\varPhi_{m'}(t,\boldsymbol{x}') , \end{equation}

where ${\mathcal {R}}_\kappa$ is called the regulator, or cutoff, matrix. Note that it has been chosen here to be proportional to $\delta (t-t')$, or equivalently independent of frequencies. This means that the selection of fluctuations is operated in space, and not in time, as in equilibrium. (The selection can in principle be operated also in time, although it poses some technical difficulties, not to violate causality (Canet, Chaté & Delamotte Reference Canet, Chaté and Delamotte2011a) and symmetries involving time – typically the Galilean invariance. A space–time cutoff was implemented only in Duclut & Delamotte (Reference Duclut and Delamotte2017) for Model A, which is a simple, purely dissipative, dynamical extension of the Ising model. In the following, we restrict ourselves to frequency-independent regulators.)

The precise form of the elements $[{\mathcal {R}}]_{\kappa,ij}$ of the cutoff matrix is not important, provided they satisfy the following requirements:

(4.3)\begin{equation} \left. \begin{aligned} & R_{\kappa}(\boldsymbol{p})\sim \kappa^{2},\quad \text{for }|\boldsymbol{p}|\lesssim \kappa, \\ & R_{\kappa}(\boldsymbol{p}) \longrightarrow 0,\quad \text{for }|\boldsymbol{p}|\gtrsim\kappa , \end{aligned} \right\} \end{equation}

where $R_k$ denotes a generic non-vanishing element of the matrix ${\mathcal {R}}\kappa$. The first constraint endows the low wavenumber modes with a large ‘mass’ $\kappa ^{2}$, such that these modes are damped, or filtered out, for their contribution in the functional integral to be suppressed. The second one ensures that the cutoff vanishes for large wavenumber modes, which are thus unaffected. Hence, only these modes are integrated over, thus achieving the progressive averaging. Moreover, $R_{\kappa =\varLambda }$ is required to be very large such that all fluctuations are frozen at the microscopic (UV) scale, and $R_{\kappa =0}$ to vanish such that all fluctuations are averaged over in this (IR) limit.

It follows that the free energy functional ${\mathcal {W}}_\kappa = \ln {\mathcal {Z}}_\kappa$ also becomes scale dependent. One defines the scale-dependent effective average action through the modified Legendre transform

(4.4)\begin{equation} \varGamma_\kappa[\varPsi] + \Delta{\mathcal{S}}_\kappa[\varPsi] = {\sup}_{{\mathcal{J}}} \left[\int_{t,\boldsymbol{x}} {\mathcal{J}}_\ell \varPsi_\ell -{\mathcal{W}}_\kappa[{\mathcal{J}}] \right]. \end{equation}

The regulator term is added to the relation in order to enforce that, at the scale $\kappa =\varLambda$, the effective average action coincides with the microscopic action $\varGamma _{\kappa =\varLambda }= {\mathcal {S}}$. For fluid dynamics, the scale $\varLambda ^{-1}$ represents a very small scale, typically a few mean free paths, where the description in term of a continuous equation becomes valid, and the ‘microscopic’ action is the NS action ${\mathcal {S}}_{NS}$. This scale is thus much smaller than the Kolmogorov scale $\eta$, and the effect of fluctuations (the renormalisation) is already important at scales $\ell \simeq \eta$, i.e. the statistical properties of the turbulence in the dissipative range are non-trivial. In the opposite limit $\kappa \to 0$ (or equivalently $\kappa \ll L^{-1}$) the standard effective action $\varGamma$, encompassing all the fluctuations, is recovered since the regulator is removed in this limit $\Delta {\mathcal {S}}_{\kappa =0}=0$. Thus, the sequence of $\varGamma _\kappa$ provides an interpolation between the microscopic action (the NS action) and the full effective action (the statistical properties of the fluid).

4.2. Exact flow equation for the effective average action

The evolution of the generating functionals ${\mathcal {W}}_\kappa$ and $\varGamma _\kappa$ with the RG scale $\kappa$ obeys an exact differential equation, which can be simply inferred from (4.1) and (4.4) since the dependence on $\kappa$ only comes from the regulator term $\Delta {\mathcal {S}}_\kappa$. The derivation is very general and can be found in standard references, e.g. § 2.3.3. of Delamotte (Reference Delamotte2012). One finds the following exact flow equation for ${\mathcal {W}}_\kappa$:

(4.5)\begin{equation} \partial_{s}{\mathcal{W}}_{\kappa}\left[{\mathcal{J}}\right] ={-}\frac{1}{2}\text{Tr}\left[\partial_{s}{\mathcal{R}}_{\kappa,mn}\left(\frac{\delta^{2}{\mathcal{W}}_{\kappa}\left[{\mathcal{J}}\right]}{\delta {\mathcal{J}}_{m}\delta {\mathcal{J}}_{n}}+\frac{\delta {\mathcal{W}}_{\kappa}\left[{\mathcal{J}}\right]}{\delta {\mathcal{J}}_{m}}\frac{\delta {\mathcal{W}}_{\kappa}\left[{\mathcal{J}}\right]}{\delta {\mathcal{J}}_{n}}\right)\right], \end{equation}

where $s\equiv \log (\kappa /\varLambda )$. This equation is very similar to the Polchinski equation (Polchinski Reference Polchinski1984). Some simple algebra then leads to the Wetterich equation for $\varGamma _\kappa$

(4.6)\begin{equation} \partial_{s}\varGamma_{\kappa}\left[\varPsi\right] = \frac{1}{2}\text{Tr}\left[\partial_{s}{\mathcal{R}}_{\kappa,mn} G_{\kappa,mn} \right]\equiv \frac{1}{2}\text{Tr}\left[\partial_{s}{\mathcal{R}}_{\kappa,mn}\left(\varGamma_{\kappa}^{\left(2\right)}\left[\varPsi\right]+{\mathcal{R}}_{\kappa}\right)_{mn}^{{-}1} \right] , \end{equation}

where $G_\kappa \equiv {\mathcal {W}}^{(2)}_\kappa$ is the propagator, and is the inverse of $(\varGamma _\kappa ^{(2)}+{\mathcal {R}}_\kappa )$ from the Legendre relation. To alleviate notations, the indices $m,n$ refer to the field indices within the multiplet, as well as other possible indices (e.g. vector component) and space–time coordinates. Accordingly, the trace includes the summation over all internal indices as well as the integration over all spatial and temporal coordinates (conforming to deWitt notation, integrals are implicit).

While (4.6) and (4.5) are exact, they are functional partial differential equations which cannot be solved exactly in general. Their functional nature implies that they encompass an infinite set of flow equations for the associated correlation functions or vertices. For instance, taking one functional derivative of (4.6) with respect to a given field and evaluating the resulting expression at a fixed background field configuration (say $\varPsi (t,\boldsymbol {x})=0$) yields the flow equation for the one-point vertex $\varGamma _\kappa ^{(1)}$. This equation depends on the three-point vertex $\varGamma _\kappa ^{(3)}$. More generally, the flow equation for the $n$-point vertex $\varGamma _\kappa ^{(n)}$ involves the vertices $\varGamma _\kappa ^{(n+1)}$ and $\varGamma _\kappa ^{(n+2)}$, such that one has to consider an infinite hierarchy of flow equations. This pertains to the very common closure problem of nonlinear systems. It means that one has to devise some approximations.

4.3. Non-perturbative approximation schemes

This part may appear very technical for readers not familiar with RG methods. Its objective is to explain the rationale underlying the approximation schemes used in the study of turbulence, which are detailed in the rest of the Perspectives article.

In the FRG context, several approximation schemes have been developed and are commonly used (Dupuis et al. Reference Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier and Wschebor2021). Of course one can implement a perturbative expansion, in any available small parameter, such as a small coupling or an infinitesimal distance to a critical dimension $\varepsilon =d - d_c$. One then retrieves results obtained from standard perturbative RG techniques. However, the key advantage of the FRG formalism is that it is suited to the implementation of non-perturbative approximation schemes. The most commonly used is the derivative expansion, which consists in expanding the effective average action $\varGamma _\kappa$ in powers of gradients and time derivatives (thus yielding an ansatz for $\varGamma _\kappa$). This is equivalent to an expansion around zero external wavenumbers $|\boldsymbol {p}_i|=0$ and frequencies $\omega _i =0$ and is thus adapted to describe the long-distance long-time properties of a system. One can in particular obtain universal properties of a system at criticality (e.g. critical exponents), but also non-universal properties, such as phase diagrams. Even though it is non-perturbative in the sense that it does not rely on an explicit small parameter, it is nonetheless controlled. It can be systematically improved, order by order (adding higher-order derivatives), and an error can be estimated at each order, using properties of the cutoff (De Polsi, Hernández-Chifflet & Wschebor Reference De Polsi, Hernández-Chifflet and Wschebor2022). The convergence and accuracy of the derivative expansion have been studied in depth for archetypal models, namely the Ising model and ${{O}}(N)$ models (Dupuis et al. Reference Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier and Wschebor2021). The outcome is that the convergence is fast, and most importantly that the results obtained for instance for the critical exponents are very precise. For the 3-D Ising model, the derivative expansion has been pushed up to the sixth order ${{O}}(\partial ^{6})$ and the results for the critical exponents compete in accuracy with the best available estimates in the literature (stemming from conformal bootstrap methods) (Balog et al. Reference Balog, Chaté, Delamotte, Marohnić and Wschebor2019). For the ${{O}}(N)$ models, the FRG results are the most accurate ones available (De Polsi et al. Reference De Polsi, Balog, Tissier and Wschebor2020, Reference De Polsi, Hernández-Chifflet and Wschebor2021).

The derivative expansion is by construction restricted to describe the zero wavenumber and frequency sector, it does not allow one to access the full space–time dependence of generic correlation functions. To overcome this limitation, one can resort to another approximation scheme, which consists in a vertex expansion. The most useful form of this approximation is called the Blaizot–Mendez–Wschebor (BMW) scheme (Blaizot, Méndez-Galain & Wschebor Reference Blaizot, Méndez-Galain and Wschebor2006, Reference Blaizot, Méndez-Galain and Wschebor2007; Benitez, Méndez-Galain & Wschebor Reference Benitez, Méndez-Galain and Wschebor2008). It lies at the basis of all the FRG studies dedicated to turbulence. The BMW scheme essentially exploits an intrinsic property of the Wetterich flow equation conveyed by the presence of the regulator. The Wetterich equation has a one-loop structure. To avoid any confusion, let us emphasise that this does not mean that it is equivalent to a one-loop perturbative expansion: the propagator entering (4.6) is the full (functional) renormalised propagator of the theory, not the bare one. This simply means that the flow equation involves only one internal, or loop, (i.e. integrated over) wavevector $\boldsymbol {q}$ and frequency $\omega$. This holds true for the flow equation of any vertex $\varGamma _\kappa ^{(n)}$, which depends on the $n$ external wavevectors $\boldsymbol {p}_i$ and frequencies $\varpi _i$, but always involves only one internal (loop) wavevector and frequency $(\omega,\boldsymbol {q})$.

The key feature of the Wetterich equation is that the internal wavevector $\boldsymbol {q}$ is controlled by the scale derivative of the regulator. Because of the requirement (4.3), $\partial _s {\mathcal {R}}_\kappa (\boldsymbol {q})$ is exponentially vanishing for $q\equiv |\boldsymbol {q}| \gtrsim \kappa$, which implies that the loop integral is effectively cut to $q \simeq \kappa$, with $\kappa$ the RG scale. This points to a specific limit where this property can be exploited efficiently, namely the large wavenumber limit. Indeed, if one considers large external wavenumbers $p_i\equiv |\boldsymbol {p}_i|\gg \kappa$, then $|\boldsymbol {q}|\ll |\boldsymbol {p}_i|$ is automatically satisfied, or otherwise stated $|\boldsymbol {q}|/|\boldsymbol {p}_i|\to 0$. Hence the limit of $|\boldsymbol {p}_i|\to \infty$ is formally equivalent to the limit $|\boldsymbol {q}|\to 0$ (because the $\boldsymbol {p}_i$ and $\boldsymbol {q}$ always come as sum or product in the propagator and vertices). Moreover, the presence of the regulator ensures that all the vertices are analytical functions of their arguments at any finite $\kappa$, and can thus be Taylor expanded (Berges et al. Reference Berges, Tetradis and Wetterich2002). The BMW approximation precisely consists in expanding the vertices entering the flow equation around $\boldsymbol {q}\simeq 0$, which can be interpreted as an expansion at large $p_i$. This expansion becomes formally exact in the limit where all $p_i\to \infty$. Besides, one has that a generic $\varGamma _\kappa ^{(n)}$ vertex with one zero wavevector can be expressed as a derivative with respect to a constant field of the corresponding vertex $\varGamma _\kappa ^{(n-1)}$ stripped of the field with zero wavevector. This allows one in the original BMW approximation scheme to close the flow equation at a given order (say for the two-point function), at the price of keeping a dependence in a background constant field $\varPsi _0$, i.e. $\bar \varGamma _\kappa ^{(2)}(\varpi,\boldsymbol {p};\varPsi _0)$, which can be cumbersome. This approximation can also in principle be improved order by order, by achieving the $\boldsymbol {q}$ expansion in the flow equation for the next order $(n+1)$ vertex instead of the $n$th one, although it becomes increasingly difficult.

In the context of non-equilibrium statistical physics, the BMW approximation scheme was implemented successfully to study the Burgers or equivalently Kardar–Parisi–Zhang equation (2.8), which is detailed in § 5. For the NS equation, the BMW approximation scheme turns out to be remarkably efficient in two respects, which will become clear in the following: technically, the symmetries and extended symmetries allow one to get rid of the background field dependence, and thus to achieve the closure exactly. Moreover, physically, the large wavenumber limit is not trivial for turbulence, which is unusual. In critical phenomena exhibiting standard scale invariance, the large wavenumbers simply decouple from the IR properties and this limit carries no independent information (see § 5).

5. A warm-up example: Burgers-KPZ equation

The Burgers equation is often considered as a toy model for classical hydrodynamics. In the inviscid limit, its solution develops shocks after a finite time even for smooth initial conditions (Bec & Khanin Reference Bec and Khanin2007). Most studies consider potential flows, for which case the (irrotational $d$-dimensional) Burgers equation exactly maps to the KPZ equation (2.8). The forcing is assumed to be a power law $D(\boldsymbol {p})\sim p^{\beta }$, which mainly injects energy at small scales (UV modes) for $\beta >0$, while it acts on large scales (IR modes) for $\beta <0$.

One is generally interested in the space–time correlations of the velocity

(5.1)\begin{equation} C^{(n)}(\tau,\ell) = \left\langle \left[\boldsymbol{v}(t+\tau,\boldsymbol{x}+\boldsymbol{\ell})-\boldsymbol{v}(t,\boldsymbol{x}) \right]^{n} \right\rangle. \end{equation}

The structure functions, which correspond to the equal-time correlations, behave as a power law in the inertial range $S_n(\ell ) \sim \ell ^{\zeta _n}$. Besides, the two-point correlation function is expected to endow a scaling form

(5.2)\begin{equation} C^{(2)}(\tau,\ell) \equiv C(\tau,\ell) = \ell^{2(\chi-1)} F(\tau/\ell^{z}), \end{equation}

where $\chi$ and $z$ are universal critical exponents called the roughness and dynamical exponent in the context of interface growth, and $F$ is a universal scaling function. The roughness exponent is simply related to $\zeta _2$ as $\zeta _2 = 2(\chi -1)$. The dynamical and roughness exponents are related in all dimensions by the exact identity $z+\chi =2$ stemming from Galilean invariance.

In one dimension $d=1$, the critical exponents are known exactly. For $\beta >0$, one has (Medina et al. Reference Medina, Hwa, Kardar and Zhang1989; Janssen, Täuber & Frey Reference Janssen, Täuber and Frey1999; Kloss et al. Reference Kloss, Canet, Delamotte and Wschebor2014a)

(5.3)\begin{equation} \chi = \max\left(\dfrac 1 2, -\dfrac \beta 3 +1\right) . \end{equation}

Moreover, shocks are overwhelmed by the forcing and there is no intermittency, i.e. $\zeta _n = n \zeta _2/2 = n(\chi -1)$ (Hayot & Jayaprakash Reference Hayot and Jayaprakash1996). For $\beta <-3$, the stationary state contains a finite density of shocks and the scaling of the velocity increments in the regime where $|\ell |$ is much smaller than the average distance between shocks but larger than the shock size can be estimated by a simple argument, yielding (Bec & Khanin Reference Bec and Khanin2007)

(5.4)\begin{equation} \zeta_n = \min\left(1, n\right) . \end{equation}

While the intermediate regime $-3<\beta <0$ is not well understood, a wealth of exact results are available for $\beta =2$, which have been obtained in the context of the KPZ equation (Corwin Reference Corwin2012). In particular, the analytical form of the scaling function $F$ in (5.2) has been determined exactly (Prähofer & Spohn Reference Prähofer and Spohn2000).

In higher dimensions $d>1$, very few are known. The most studied case corresponds to the KPZ equation $\beta =2$. In contrast with $d=1$ where the interface always roughens, in dimensions $d>2$, the KPZ equation exhibits a (non-equilibrium) continuous phase transition between a smooth phase and a rough phase, depending on the amplitude $\lambda$ of the nonlinearity in (2.8). For $\lambda <\lambda _c$, the interface remains smooth, it is then described by the Edwards–Wilkinson equation which is the linear, non-interacting, $\lambda =0$, version of KPZ, and which has a simple diffusive behaviour with $z=2$ and $\chi =(2-d)/2$. For $\lambda >\lambda _c$, the nonlinearity is dominant and the interface becomes rough, with $\chi >0$. This is the KPZ phase. Contrarily to $d=1$, no exact result has been obtained for $d\neq 1$, and the critical exponents are known only numerically (Pagnani & Parisi Reference Pagnani and Parisi2015).

Since the early days of the KPZ equation, perturbative RG techniques (usually referred to as dynamical RG) have been applied to determine the critical exponents (Kardar et al. Reference Kardar, Parisi and Zhang1986; Medina et al. Reference Medina, Hwa, Kardar and Zhang1989; Janssen et al. Reference Janssen, Täuber and Frey1999). However, the KPZ equation constitutes a striking example where the perturbative RG flow equations are known to all orders in perturbation theory, but nonetheless fail for $d\geq 2$ to find the strong-coupling fixed point expected to govern the KPZ rough phase (Wiese Reference Wiese1998). In contrast, the FRG framework allows one to access this fixed point in all dimensions even at the lowest order of the derivative expansion (Canet Reference Canet2005; Canet et al. Reference Canet, Chaté, Delamotte and Wschebor2010). This fixed point turns out to be genuinely non-perturbative, i.e. not connected to the Edwards–Wilkinson fixed point (which is the point around which perturbative expansions are performed) in any dimension, hence explaining the failure of perturbation theory, to all orders. We now briefly review these results.

5.1. The FRG for the Burgers–KPZ equation

The correlation function (5.2) can be computed from the two-point functions $\varGamma ^{(2)}$. The general FRG flow equation for these functions is obtained by differentiating twice the exact flow equation (4.6) for the effective average action $\varGamma _\kappa$. Assuming translational invariance in space and time, it can be evaluated at zero fields and then reads

(5.5)\begin{align} & \partial_\kappa \bar\varGamma^{(2)}_{\kappa,\ell m}(\varpi,\boldsymbol{p})= {\rm Tr} \int_{\omega,\boldsymbol{q}} \partial_\kappa {\mathcal{R}}_\kappa(\boldsymbol{q}) \boldsymbol{\cdot} \bar G_\kappa(\omega,\boldsymbol{q}) \boldsymbol{\cdot} \left(-\frac{1}{2} \bar\varGamma^{(4)}_{\kappa,\ell m}(\varpi,\boldsymbol{p},-\varpi,-\boldsymbol{p},\omega,\boldsymbol{q})\right. \nonumber\\ &\left.\qquad + \bar\varGamma^{(3)}_{\kappa,\ell}(\varpi,\boldsymbol{p},\omega,\boldsymbol{q}) \boldsymbol{\cdot} \bar G_\kappa(\varpi+\omega,\boldsymbol{p}+\boldsymbol{q}) \boldsymbol{\cdot} \bar\varGamma^{(3)}_{\kappa,m}(-\varpi,-\boldsymbol{p},\varpi+\omega,\boldsymbol{p}+\boldsymbol{q}) \vphantom{\frac{1}{2}}\right) \boldsymbol{\cdot} G_\kappa(\omega,\boldsymbol{q}), \end{align}

where the last arguments of the $\bar \varGamma _\kappa ^{(n)}$ are implicit since they are determined by frequency and wavevector conservation according to (3.19). We used a matrix notation, where only the external field index $\ell$ and $m$ are specified, i.e. $\bar \varGamma ^{(3)}_{\kappa,\ell }$ is the $2\times 2$ matrix of all three-point vertices with one leg fixed at $(\ell,\varpi,\boldsymbol {p})$ and the other two spanning all fields, and similarly for $\bar \varGamma ^{(4)}_{\kappa,\ell m}$. It can be represented diagrammatically as in figure 1. In order to make explicit calculations, one has to make some approximation. Here, we resort to an ansatz for the effective average action $\varGamma _\kappa$, which then enables one to compute all the $n$-point functions entering the flow equation. A crucial point for the choice of this ansatz is to preserve the symmetries of the model. The fundamental symmetry of the Burgers equation is the Galilean invariance. As a consequence, the $n$-point vertices $\varGamma^{(n)}$ are related by similar identities as (3.31) (simply removing the pressure fields). Thus, a satisfactory ansatz for the Burgers effective average action should automatically satisfy all these identities. To devise such an ansatz is not a simple task in general. However, the construction can be understood in a rather intuitive way.

Let us define a function $f(t,\boldsymbol {x})$ as a scalar density under the Galilean transformation (3.25) if its infinitesimal variation under this transformation is $\delta f(t,\boldsymbol {x}) = \varepsilon _\alpha (t)\partial _\alpha f(t,\boldsymbol {x})$ – i.e. it varies only due to the change of coordinates. This then implies that $\int _{\boldsymbol {x}} f(t,\boldsymbol {x})$ is invariant under a Galilean transformation. According to (3.25), $\bar {\boldsymbol {v}}$ is a scalar density, while $\boldsymbol {v}$ is not. However, one can build two scalar densities from it, which are $\partial _\alpha \boldsymbol {v}$ and the Lagrangian time derivative $D_t \boldsymbol {v} \equiv \partial _t \boldsymbol {v}+ {{v}}_\alpha \partial _\alpha \boldsymbol {v}$. One can then easily show that combining scalars together through sums and products preserves the scalar density property, as well as applying a gradient. While applying a time derivative $\partial _t$ spoils the scalar density property, the Lagrangian time derivative preserves it, which identifies $D_t$ as the covariant time derivative for Galilean transformation. These rules allow one to construct an ansatz which is manifestly invariant under the Galilean transformation.

The most advanced approximation, denoted SOA, which has been implemented so far for the Burgers–KPZ equation consists in truncating the effective average action $\varGamma _\kappa$ at second order in the response field, and thus neglecting higher-order terms in this field which could in principle be generated by the RG flow. This means that the noise probability distribution is kept as Gaussian as in the original Langevin description, which sounds reasonable. The most general ansatz at this order compatible with the symmetries of the Burgers equation, i.e. using the previous construction, reads

(5.6)\begin{equation} \varGamma_\kappa[\boldsymbol{u},\bar{\boldsymbol{u}}] = \int_{t,\boldsymbol{x}}\{\bar u_\alpha f_{\kappa}^{\rho}(D_t,\boldsymbol{\nabla}) D_t u_\alpha -\bar u_{\alpha} f^{\nu}_{\kappa}(D_t,\boldsymbol{\nabla}) \nabla^{2} u_\alpha -\bar u_\alpha f^{D}_{\kappa}(D_t,\boldsymbol{\nabla}) \bar u_\alpha\}. \end{equation}

It is parametrised by three scale-dependent renormalisation functions $f^{\nu }_{\kappa }$, $f^{D}_{\kappa }$ and $f^{\rho }_{\kappa }$, which are arbitrary functions of the covariant operators $\boldsymbol {\nabla }$ and $D_t$, beside depending on the RG scale $\kappa$. (Note that $f^{\nu }_{\kappa }$, $f^{D}_{\kappa }$ and $f^{\rho }_{\kappa }$ are scalar functions because they correspond to the longitudinal parts of $f^{\nu }_{\kappa,\alpha \beta }$, $f^{D}_{\kappa,\alpha \beta }$ and $f^{\rho }_{\kappa,\alpha \beta }$. The transverse parts vanish since the flow is potential.) Their initial condition at $\kappa =\varLambda$ is $f^{\nu }_\varLambda =\nu$, $f^{D}_\varLambda =(D_{\alpha \beta })^{\parallel }$ (longitudinal part of $D_{\alpha \beta }$) and $f^{\rho }_\varLambda =1$ for which the initial Burgers–KPZ action (3.20) is recovered. The ansatz (5.6) encompasses the most general dependence in wavenumbers and frequencies, and also in velocity fields, of the two-point functions compatible with Galilean invariance. Indeed, arbitrary powers of the velocity field $\boldsymbol {u}$ are included through the operator $D_t$. Thus $\varGamma _\kappa$ is truncated only in the response field, not in the field itself.

The choice of an ansatz allows one to compute explicitly all the vertices $\varGamma _\kappa ^{(n)}$, their expression can be found in Canet et al. (Reference Canet, Chaté, Delamotte and Wschebor2011b). For example, it yields for the two-point functions in Fourier space

(5.7)\begin{equation} \left. \begin{aligned} \bar{\varGamma}_\kappa^{(1,1)}(\omega,\boldsymbol{p}) & = {\rm i}\omega f_\kappa^{\rho}\left(\omega, \boldsymbol{p}\right) + \boldsymbol{p}\,^{2} f_\kappa^{\nu}(\omega,\boldsymbol{p}) \\ \bar{\varGamma}_\kappa^{(0,2)}(\omega,\boldsymbol{p}) & ={-}2 f_\kappa^{D}(\omega,\boldsymbol{p}). \end{aligned} \right\} \end{equation}

The flow equations for the three functions $f^{\nu }_\kappa$, $f^{D}_\kappa$ and $f^{\rho }_\kappa$ can then be computed from the flow equations (5.5). More details on the calculation, in particular on the choice of the regulator, are provided in § 6 for NS, and can be found for Burgers–KPZ in e.g. (Canet et al. Reference Canet, Chaté, Delamotte and Wschebor2011b).

5.2. Scaling dimensions

With the aim of analysing the fixed-point structure, one defines dimensionless and renormalised quantities. First, one introduces scale-dependent coefficients $\nu _\kappa \equiv f_\kappa ^{\nu }(0,0)$ and $D_\kappa \equiv f_\kappa ^{D}(0,0)$ (equivalently $D_\kappa$ is defined from $f_\kappa ^{D}$ evaluated at a finite $\boldsymbol {p}$ if $f_\kappa ^{D}$ is non-analytic at $p=0$) which identify at the microscopic scale $\kappa =\varLambda$ with the parameters $\nu$ and $D$ of the action (3.20). (Note that the scaling dimension of $f_\kappa ^{\rho }$ is fixed to 1 by the symmetries so there is no need to introduce a coefficient $\rho _\kappa$ associated with this function.) They are associated with scale-dependent anomalous dimensions as

(5.8a,b)\begin{equation} \eta_\kappa^{D} ={-}\partial_s \ln D_\kappa ,\quad \eta^{\nu}_\kappa ={-}\partial_s \ln \nu_\kappa , \end{equation}

where $s=\ln (\kappa /\varLambda )$ is the RG ‘time’ and $\partial _s = \kappa \partial _\kappa$. Indeed, one expects that if a fixed point is reached, these coefficients behave as power laws $D_\kappa \sim \kappa ^{-\eta _*^{D}}$ and $\nu _\kappa \sim \kappa ^{-\eta _*^{\nu }}$ where the $*$ denotes fixed-point quantities. The physical critical exponents can then be deduced from $\eta ^{D}_*$ and $\eta ^{\nu }_*$. For this, let us determine the scaling dimensions of the fields from (5.6). The three functions $f^{\nu }_\kappa$, $f^{D}_\kappa$ and $f^{\rho }_\kappa$ have respective scaling dimensions $\nu _\kappa$, $D_\kappa$ and $\rho _\kappa \equiv 1$ by definition of these coefficients. Since $\varGamma _\kappa$ has no dimension, one deduces from the term $\propto \partial _t$ in $\varGamma _\kappa$ that $[u_\alpha \bar u_\alpha ] = \kappa ^{d}$ and from the viscosity term that $[u_\alpha \bar u_\alpha ]=\kappa ^{d} \omega \kappa ^{-2}\nu _\kappa ^{-1}$. Equating the two terms, this implies that the scaling dimension of the frequency is $[\omega ] = \nu _\kappa \kappa ^{2} \equiv \kappa ^{z}$, which defines the dynamical critical exponent as $z=2-\eta ^{\nu }_*$. The scaling dimensions of the fields can then be deduced from the forcing term of $\varGamma _\kappa$ as

(5.9a,b)\begin{equation} [u_\alpha] = (\kappa^{d-2}D_\kappa \nu_\kappa^{{-}1})^{1/2} ,\quad [\bar u_\alpha] = (\kappa^{d+2}\nu_\kappa D_\kappa^{{-}1})^{1/2} . \end{equation}

This implies in particular that the roughness critical exponent $\chi$ is related to $\eta _*^{\nu }$ and $\eta _*^{D}$ as $\chi = (2-d-\eta _*^{\nu }+\eta _*^{D})/2$.

To introduce dimensionless quantities, denoted with a hat symbol, wavevectors are measured in units of $\kappa$, e.g. $\boldsymbol {p} =\kappa \hat {\boldsymbol {p}}$ and frequencies in units of $\nu _\kappa \kappa ^{2}$, e.g. $\omega = \nu _\kappa \kappa ^{2} \hat {\omega }$. Dimensionless fields are defined as $\hat {\boldsymbol {u}} =(\kappa ^{d-2}D_\kappa \nu _\kappa ^{-1})^{-1/2} \boldsymbol {u}$, $\hat {\bar {\boldsymbol {u}}} = (\kappa ^{d+2}\nu _\kappa D_\kappa ^{-1})^{-1/2}\bar {\boldsymbol {u}}$, and dimensionless functions as

(5.10ac)\begin{align} \hat{f}_\kappa^{\nu}(\hat\omega,\hat{\boldsymbol{p}})\equiv\frac{1}{\nu_\kappa} f_\kappa^{\nu}\left(\frac{\omega}{\nu_\kappa \kappa^{2}},\frac{\boldsymbol{p}}{\kappa}\right),\quad \hat{f}_\kappa^{D}(\hat\omega,\hat{\boldsymbol{p}}) \equiv\frac{1}{D_\kappa} f_\kappa^{D}\left(\frac{\omega}{\nu_\kappa \kappa^{2}},\frac{\boldsymbol{p}}{\kappa}\right),\quad \hat{f}_\kappa^{\rho}(\hat\omega,\hat{\boldsymbol{p}}) \equiv f_\kappa^{\rho}\left(\frac{\omega}{\nu_\kappa \kappa^{2}},\frac{\boldsymbol{p}}{\kappa}\right). \end{align}

The non-dimensionalisation of the fields introduces a scaling dimension in front of the advection term. We define the corresponding dimensionless coupling as $\hat {\lambda }_\kappa = (\kappa ^{d-4}D_\kappa \nu _\kappa ^{-3})^{1/2}\lambda$ (where $\lambda$ is the KPZ parameter, $\lambda =1$ for the Burgers equation). The Ward identity for Galilean symmetry yields that $\lambda$ is not renormalised, i.e. it stays constant throughout the RG flow, or otherwise stated its flow is zero $\partial _s \lambda =0$. One thus obtains for the flow of $\hat {\lambda }_\kappa$

(5.11)\begin{equation} \partial_s \hat{\lambda}_\kappa = \frac{\hat\lambda_\kappa}{2} (d -4+ 3 \eta_\kappa^{\nu} -\eta_\kappa^{D}) . \end{equation}

One can deduce from this equation that if a non-trivial fixed point exists ($\hat \lambda _*\neq 0$), then the fixed-point values of the anomalous dimensions satisfy the exact relation

(5.12)\begin{equation} d -4+ 3 \eta_*^{\nu} -\eta_*^{D}=0 . \end{equation}

Using the definitions of the critical exponents, this relation writes $\chi +z=2$ in all dimensions, which is the exact identity mentioned in the introduction of this section.

5.3. Results in $d=1$

Let us focus on the case $\beta =2$, i.e. $D_{\alpha \beta }(\boldsymbol {p})\equiv D p_\alpha p_\beta$ in (3.20), which corresponds to the KPZ equation, because exact results are available for this case in one dimension. In $d=1$, there exists an additional discrete time-reversal symmetry which greatly simplifies the problem. This is one of the reasons why exact results could be obtained in this dimension only. In particular, one can show that it implies a particular form of fluctuation–dissipation theorem which writes for the two-point functions (Frey & Täuber Reference Frey and Täuber1994; Canet et al. Reference Canet, Chaté, Delamotte and Wschebor2011b)

(5.13)\begin{equation} 2{Re}\bar\varGamma_\kappa^{(1,1)}(\omega,\boldsymbol{p}) ={-}\frac{\nu}{D}p^{2}\bar\varGamma_\kappa^{(0,2)}(\omega,\boldsymbol{p}), \end{equation}

and in turn entails that $D_\kappa =\nu _\kappa$, $f_\kappa ^{D} = f_\kappa ^{\nu }\equiv f_\kappa$ and $f_\kappa ^{\rho }\equiv 1$. (Note that the viscosity term in (5.6) has to be symmetrised to exactly preserve the Ward identities ensuing from the time-reversal symmetry for all the $n$-point vertices, see Canet et al. (Reference Canet, Chaté, Delamotte and Wschebor2011b).) Thus there remains only one renormalisation function $f_\kappa$ and one anomalous dimension $\eta _\kappa ^{\nu }=\eta _\kappa ^{D}\equiv \eta _\kappa$ to compute in $d=1$. This implies that the fixed-point value of the latter is fixed exactly by (5.12) to $\eta _*=1/2$, which yields the well-known values $\chi =1/2$ and $z=3/2$. The flow equation for the associated dimensionless function $\hat {f}_\kappa$ reads

(5.14)\begin{equation} \partial_s \hat{f}_\kappa(\hat{\omega},\hat{\boldsymbol{p}}) = [\eta_\kappa +(2-\eta_\kappa) \hat{\omega} \;\partial_{\hat{\omega}} + \hat{\boldsymbol{p}} \;\partial_{\hat{\boldsymbol{p}}}]\hat{f}_\kappa(\hat{\omega},\hat{\boldsymbol{p}}) +\frac{1}{D_\kappa} \partial_s f_\kappa(\omega,\boldsymbol{p}), \end{equation}

where the first term in square bracket comes from the non-dimensionalisation (5.10ac) and the last term corresponds to the nonlinear loop term calculated from the flow equations (5.5), which has to be expressed in terms of the dimensionless (hat) quantities.

This flow equation can be integrated numerically, from the initial condition ${\hat {f}_{\varLambda }(\hat \omega,\hat {\boldsymbol{p}})=1}$ at scale $\kappa =\varLambda$. This has been performed in Canet et al. (Reference Canet, Chaté, Delamotte and Wschebor2011b), to which we refer the reader for details. One obtains that the flow reaches a fixed point, where all quantities stop evolving. The function $\hat {f}_\kappa$ converges to a fixed form $\hat {f}_*(\hat \omega,\hat {\boldsymbol {p}})$, which behaves as a power law at large $\boldsymbol {p}$ and large $\omega$ as

(5.15a,b)\begin{equation} \hat{f}_{\kappa}(\hat\omega,\hat{\boldsymbol{p}}) \stackrel{|\hat{\boldsymbol{p}}|\gg 1}{\sim} |\boldsymbol{p}|^{-\eta_*},\quad \hat{f}_{\kappa}(\hat\omega,\hat{\boldsymbol{p}}) \stackrel{\omega\gg 1}{\sim} \omega^{-\eta_*/(2-\eta_*)}. \end{equation}

In fact, this is expected from a very important property of the flow equation called decoupling. Decoupling means that the loop contribution ${\partial _s f_{\kappa }}/{D_\kappa }$ in the flow equation (5.14) becomes negligible in the limit of large wavenumbers and/or frequencies compared with the linear terms. Intuitively, a large wavenumber is equivalent to a large mass, and degrees of freedom with a large mass are damped and do not contribute in the dynamics at large scales. This means that the IR (effective) properties are not affected by the UV (microscopic) details, and this pertains to the mechanism for universality. Moreover, one can prove that the decoupling property entails scale invariance. Indeed, if the loop contribution $\partial _s \hat f_{\kappa }$ is negligible compared with the linear terms, one can readily prove that the general solution of (5.14) takes the scaling form

(5.16)\begin{equation} \hat{f}^{*}(\hat{\omega},\hat{\boldsymbol{p}}) = \frac{1}{|\hat{\boldsymbol{p}}|^{\eta_*}} \hat{\zeta}\left(\frac{\hat{\omega}}{|\hat{\boldsymbol{p}}|^{z}}\right), \end{equation}

where $\hat {\zeta }$ is a universal scaling function, which can be determined by explicitly integrating the full flow equation (5.14). The physical correlation function (5.2) can be deduced from the ansatz as

(5.17)\begin{equation} C(\omega,\boldsymbol{p}) ={-} \frac{\varGamma^{(0,2)}(\omega,\boldsymbol{p})}{|\varGamma^{(1,1)}(\omega,\boldsymbol{p})|^{2}} = \frac{2 f(\omega,\boldsymbol{p})}{\omega^{2} + p^{4} f(\omega,\boldsymbol{p})^{2}}. \end{equation}

Replacing the function $f$ by its fixed-point form (5.16) and using the value $\eta _*=1/2$, one obtains

(5.18)\begin{equation} \hat C(\hat\omega,\hat{\boldsymbol{p}}) = \frac{2}{\hat p^{7/2}} \frac{ \hat{\zeta} \left(\dfrac{\hat{\omega}}{\hat{p}^{3/2}}\right)}{{\hat{\omega}^{2}}/{\hat{p}^{3}}+\hat{\zeta}^{2} \left(\dfrac{\hat{\omega}}{\hat{p}^{3/2}}\right)} \equiv \frac{2}{\hat p^{7/2}} \mathring{F}\left(\frac{\hat\omega}{\hat p^{3/2}}\right). \end{equation}

The physical (dimensional) correlation function $C(\omega,\boldsymbol {p})$ takes the exact same form up to normalisation constants which are fixed in terms of the KPZ parameters $\nu$, $D$ and $\lambda$ and the fixed-point value of the dimensionless coupling $\hat \lambda _*$.

In the context of the KPZ equation, it is a very important result. First, this proves the existence of generic scaling, i.e. that the KPZ interface always becomes critical (rough) in $d=1$, and so that its correlations are indeed described by the universal scaling form (5.2). Second, the associated scaling functions have been computed exactly in Prähofer & Spohn (Reference Prähofer and Spohn2000), and can be compared with the FRG result. In particular, the exactly known function, denoted $f(y)$, and its Fourier transform, denoted $\tilde {f}(k)$, are related to the FRG function $\mathring {F}$ by the integral relations

(5.19a,b)\begin{equation} \tilde{f}(k) = \int_0^{\infty} \frac{{\rm d}\tau}{\rm \pi} \cos(\tau k^{3/2}) \mathring{F}(\tau) ,\quad f(y) = \int_0^{\infty}\frac{{\rm d}k}{\rm \pi} \cos(k y )\tilde{f}(k). \end{equation}

The functions $f(y)$ and $\tilde {f}(k)$ computed from the FRG approach are displayed in figure 2 together with the exact results. Let us emphasise that there are no fitting parameters. The FRG functions match with extreme accuracy the exact ones. Interestingly, the function $\tilde {f}(k)$ is studied in detail in Prähofer & Spohn (Reference Prähofer and Spohn2004), which show that this function first monotonically decreases to vanish at $k_0\simeq 4.4$, then exhibits a negative dip, after which it decays to zero with a stretched exponential tail, over which are superimposed tiny oscillations around zero, only apparent on a logarithmic scale (inset of figure 2). The FRG function reproduces all these features. Regarding the tiny magnitude over which they develop, this agreement is remarkable. The Burgers equation with other values of $\beta >0$ has been studied in Kloss et al. (Reference Kloss, Canet, Delamotte and Wschebor2014a), which confirms the result (5.3).

Figure 2. Scaling function $f(y)$ (a) and $\tilde f(k)$ (b) for the 1-D Burgers–KPZ equation from FRG compared with the exact result from Prähofer & Spohn (Reference Prähofer and Spohn2000).

5.4. Results in $d>1$

In dimensions $d>1$, the Burgers equation has also been studied only for $\beta >0$, which is equivalent to the KPZ equation with long-range noise. As explained at the beginning of the section, for $\beta =2$, the KPZ rough phase is controlled by a strong-coupling fixed point, and perturbative RG fails to all orders to find it. Moreover, none of the mathematical methods used to derive exact results in $d=1$ can be extended to non-integrable cases, which include $d\neq 1$. In contrast, the FRG can be straightforwardly applied to any dimension, and thereby offers the only controlled analytical approach to study this problem, and it has been used in several works (Kloss, Canet & Wschebor Reference Kloss, Canet and Wschebor2012; Kloss et al. Reference Kloss, Canet, Delamotte and Wschebor2014a; Kloss, Canet & Wschebor Reference Kloss, Canet and Wschebor2014b; Squizzato & Canet Reference Squizzato and Canet2019). We will not review all these results here since they are mainly concern with KPZ. Perhaps the most noteworthy result is that the FRG provided the critical exponents and scaling functions in the physical dimensions $d=2,3$, as well as other universal quantities, such as universal amplitude ratios. The estimates for these amplitude ratios in $d=2,3$ were later confirmed in large-scale numerical simulations within less than 1 % error in Halpin-Healy (Reference Halpin-Healy2013a,Reference Halpin-Healyb). The whole scaling function in $d=2$ was computed in numerical simulations of the non-equilibrium Gross–Pitaevskii equation, which surprisingly falls into the KPZ universality class for certain parameters (Deligiannis et al. Reference Deligiannis, Fontaine, Squizzato, Richard, Ravets, Bloch, Minguzzi and Canet2022). The numerical result agrees very precisely with the FRG result, as shown in figure 3.

Figure 3. Scaling function for the 2-D Burgers–KPZ equation from FRG compared with numerical simulations from Deligiannis et al. (Reference Deligiannis, Fontaine, Squizzato, Richard, Ravets, Bloch, Minguzzi and Canet2022).

For other values of $\beta$, the results (5.3) were extended to $d>1$, yielding

(5.20)\begin{equation} \chi = \max\left(\dfrac 1 3(4-d-\beta_t(d), -\dfrac 1 3(4-d-\beta) +1\right) , \end{equation}

where $\beta _t(1)=3/2$ and monotonically decreases with $d$. The complete phase diagram of the problem has also been determined in Kloss et al. (Reference Kloss, Canet, Delamotte and Wschebor2014a).

The values $\beta >0$ correspond to a forcing mainly exerted at small scales, where it rather plays the role of a microscopic noise, which can be interpreted as a thermal noise for $\beta =2$. This then describes a fluid at rest submitted to thermal fluctuations, as studied in Forster et al. (Reference Forster, Nelson and Stephen1977). The values $\beta <0$ correspond to a large-scale forcing relevant for turbulence. This case has not been studied yet using FRG but it certainly deserves to be addressed in future works. In particular, it would be interesting to investigate whether the decoupling property breaks down for $\beta <0$, as occurs for NS, which is presented in the next section.

6. Fixed point for NS turbulence

As mentioned in § 2, perturbative RG approaches to turbulence have been hindered by the need to define a small parameter. The introduction of a forcing with power-law correlations $\propto p^{4-d-\varepsilon }$ leads to a fixed point with an $\varepsilon$-dependent energy spectrum, which is not easily linked to the expected Kolmogorov one. In this respect, the first achievement of FRG was to find the fixed point corresponding to fully developed turbulence generated by a physical forcing concentrated at the integral scale. As explained in § 4.3, the crux is that one can devise in the FRG framework approximation schemes which are not based on a small parameter, thereby circumventing the difficulties encountered by perturbative RG. This fixed point was first obtained in a pioneering work by Tomassini (Tomassini Reference Tomassini1997), which was largely overlooked at the time. He developed an approximation close in spirit to the BMW approximation, although it was not yet invented at the time. In the meantime, the FRG framework was successfully developed to study the Burgers–KPZ equation, as stressed in § 5. Inspired by the KPZ example, the stochastic NS equation was revisited using similar FRG approximations in Mejía-Monasterio & Muratore-Ginanneschi (Reference Mejía-Monasterio and Muratore-Ginanneschi2012) and Canet et al. (Reference Canet, Delamotte and Wschebor2016). We now present the resulting fixed point describing fully developed turbulence.

To show the existence of the fixed point and characterise the associated energy spectrum, one can focus on the two-point correlation functions. We follow in this section the same strategy as for the Burgers problem, that is, we resort to an ansatz for the effective average action $\varGamma _\kappa$ to close the flow equations. It is clear that it is an approximation, and we establish the existence of the fixed point within this approximation. However, as manifest in § 5, it constitutes a very reliable approximation, built and constrained from the symmetries, which leads to extremely accurate results in the case of the Burgers equation. It is certainly reliable enough to prove the very existence of the fixed point. Of course, the quantitative estimates associated with this fixed point could be systematically improved by implementing successive orders of the approximation scheme.

6.1. Ansatz for the effective average action

One can first infer the general structure of $\varGamma _\kappa$ for NS equations stemming from the symmetry constraints analysed in § 3.3 and Appendix A. On can show that it endows the following form:

(6.1)\begin{equation} \varGamma_\kappa[\boldsymbol{u},\bar{\boldsymbol{u}},p,\bar p] = \int_{t,\boldsymbol{x}}\left[ \bar u_\alpha \left(\partial_t u_\alpha+ u_\beta \partial_\beta u_\alpha +\frac{\partial_\alpha {\rm \pi}}{\rho}\right) +\bar {\rm \pi}\partial_\alpha u_\alpha\right] +\tilde \varGamma_\kappa[\boldsymbol{u},\bar{\boldsymbol{u}}], \end{equation}

where the explicit terms in square brackets are not renormalised, i.e. they remain unchanged throughout the RG flow, and are thus identical to the corresponding terms in the NS action. The functional $\tilde \varGamma _\kappa$ is renormalised, but it has to be invariant under all the extended symmetries of the NS action since they are preserved by the RG flow.

Let us now comment on the choice of the regulator. It should be quadratic in the fields, satisfy the general constraints (4.3), and preserve the symmetries. A suitable choice is

(6.2)\begin{align} \Delta {\mathcal{S}}_\kappa[\boldsymbol{u},\bar{\boldsymbol{u}}] ={-}\int_{t,\boldsymbol{x},\boldsymbol{x}'} \left\{\bar u_\alpha(t,\boldsymbol{x}) N_{\kappa,\alpha \beta}(|\boldsymbol{x}-\boldsymbol{x}'|)\bar u_\beta(t,\boldsymbol{x}')+\bar u_\alpha(t,\boldsymbol{x}) R_{\kappa,{\alpha \beta}}(|\boldsymbol{x}-\boldsymbol{x}'|) u_\beta(t,\boldsymbol{x}') \right\} . \end{align}

The first term is simply the original forcing term, promoted to a regulator by replacing the integral scale $L$ by the RG scale $\kappa ^{-1}$, since it naturally satisfies all the requirements. The forcing thus builds up with the RG flow, and the physical forcing scale is restored when $\kappa = L^{-1}$. The additional $R_\kappa$ term can be interpreted as an Eckman friction term in the NS equation. Its presence is fundamental in $d=2$ to damp the energy transfer towards the large scales. Within the FRG formalism, the energy pile-up at large scales manifests itself as an IR divergence in the flow equation when the $R_\kappa$ term is absent, which is removed by its presence. This term is thus mandatory to properly regularise the RG flow in $d=2$. Its effect is to introduce an effective energy dissipation at the boundary of the effective volume $\kappa ^{-d}$. This form of dissipation is negligible in $d=3$ compared with the dissipation at the Kolmogorov scale.

Introducing a scale-dependent forcing amplitude $D_\kappa$ and a scale-dependent viscosity $\nu _\kappa$, the two regulator terms can be parametrised as

(6.3a,b)\begin{equation} N_{\kappa,\alpha \beta}(\boldsymbol{q})=\delta_{\alpha\beta} D_ \kappa \hat{n}(\boldsymbol{q}/\kappa),\quad R_{\kappa,\alpha \beta}(\boldsymbol{q})=\delta_{\alpha\beta} \nu_ \kappa \boldsymbol{q}^{2}\hat{r}(\boldsymbol{q}/\kappa). \end{equation}

They are chosen diagonal in components without loss of generality since the propagators are transverse due to incompressibility, and thus the component $\propto q_\alpha q_\beta$ plays no role in the flow equations. The general structure of the propagator matrix $\bar G_\kappa$, i.e. the inverse of the Hessian of $\varGamma _\kappa +{\Delta {\mathcal {S}}_\kappa }$ is determined in Appendix B. One can show in particular that the propagator in the velocity sector is purely transverse

(6.4a,b)\begin{equation} \bar G_{\kappa,\alpha\beta}(\omega,\boldsymbol{q}) = P^{{\perp}}_{\alpha\beta}(\boldsymbol{q}) \bar G_{\kappa,\perp}(\omega,\boldsymbol{q}) , \quad P^{{\perp}}_{\alpha\beta}(\boldsymbol{q}) = \delta_{\alpha\beta} -\frac{q_\alpha q_\beta}{q^{2}} . \end{equation}

So far we have just given in (6.1) the general structure of $\varGamma _\kappa$ stemming from symmetry constraints. Now, one needs to make an approximation in order to solve the flow equations. For this, we devise an ansatz for $\tilde \varGamma$, which is very similar to the one considered for the Burgers problem. It is built to automatically preserve the Galilean symmetry. For NS, a lower-order approximation than the SOA one (5.6) was implemented. It is called LO (leading order) and was first introduced in the context of Burgers–KPZ in Canet et al. (Reference Canet, Chaté, Delamotte and Wschebor2010). The LO approximation consists in keeping the most general wavevector, but not frequency, dependence for the two-point functions. The corresponding ansatz reads

(6.5)\begin{equation} \tilde \varGamma_\kappa[\boldsymbol{u},\bar{\boldsymbol{u}}] = \int_{t,\boldsymbol{x},\boldsymbol{x}'}\{\bar u_\alpha(t,\boldsymbol{x}) f^{\nu}_{\kappa,\alpha \beta}(\boldsymbol{x}-\boldsymbol{x}') u_\beta(t,\boldsymbol{x}') -\bar u_\alpha(t,\boldsymbol{x}) f^{D}_{\kappa,\alpha \beta}(\boldsymbol{x}-\boldsymbol{x}') \bar u_\beta(t,\boldsymbol{x}')\}. \end{equation}

Compared with (5.6), the functions $f^{\nu }_\kappa$ and $f^{D}_\kappa$ no longer depend on the covariant operator $D_t$, which amounts to neglecting the frequency dependence, but also the dependence in the field $\boldsymbol {u}$. For this reason, all higher-order vertices apart from the one present in the original NS action vanish. This means that the renormalisation of multi-point interactions is neglected in this ansatz, which is a rather simple approximation. However, it is sufficient for the Burgers–KPZ problem to find the fixed point in all dimensions. At LO, the flow is projected onto the two renormalisation functions $f_\kappa ^{\nu }$ and $f_\kappa ^{D}$ which can be interpreted as an effective viscosity and an effective forcing.

The initial condition of the flow at scale $\kappa =\varLambda$ corresponds to

(6.6a,b)\begin{equation} f^{D}_{\varLambda,\alpha \beta}(\boldsymbol{x}-\boldsymbol{x}')=0 ,\quad f^{\nu}_{\varLambda,\alpha \beta}(\boldsymbol{x}-\boldsymbol{x}')={-}\nu \delta_{\alpha \beta}\nabla_x^{2} \delta^{(d)}(\boldsymbol{x}-\boldsymbol{x}') \end{equation}

such that one recovers the original NS action (3.22). (Note that the forcing has been incorporated in the regulator part $\Delta {\mathcal {S}}_\kappa$ which is why the initial condition of $f^{D}_{\varLambda,\alpha \beta }$ is zero.)

The calculation of the two-point functions from the LO ansatz is straightforward. At vanishing fields and in Fourier space, one obtains

(6.7)\begin{equation} \left. \begin{aligned} \tilde\varGamma_{\kappa,\alpha\beta}^{(1,1)}(\omega,\boldsymbol{p}) & =f^{\nu}_{\kappa,\alpha\beta}(\boldsymbol{p}),\\ \tilde\varGamma_{\kappa,\alpha\beta}^{(0,2)}(\omega,\boldsymbol{p}) & ={-}2f^{D}_{\kappa,\alpha\beta}(\boldsymbol{p}). \end{aligned} \right\} \end{equation}

Within the LO approximation, the only non-zero vertex function is the one present in the NS action, which reads in Fourier space

(6.8)\begin{equation} \bar \varGamma_{\kappa,\alpha\beta\gamma}^{(2,1)}(\omega_1,\boldsymbol{p}_1,\omega_2,\boldsymbol{p}_2) ={-}{\rm i} (p_2^{\alpha} \delta_{\beta\gamma} + p_1^{\beta} \delta_{\alpha\gamma}). \end{equation}

One can then compute the flow equations for the two-point functions $\tilde \varGamma ^{(1,1)}_{\kappa,\alpha \beta }$ and $\tilde \varGamma ^{(0,2)}_{\kappa,\alpha \beta }$ from (5.5). They are purely transverse, and one can deduce from them the flow equations for $f_{\kappa,\perp }^{\nu }\equiv P_{\alpha \beta }^{\perp } f^{\nu }_{\kappa,\alpha \beta }$ and $f_{\kappa,\perp }^{D}\equiv P_{\alpha \beta }^{\perp } f^{D}_{\kappa,\alpha \beta }$, see Appendix B for details and their explicit expressions.

6.2. Stationarity

Since we are interested in the existence of a fixed point, it is convenient to non-dimensionalise all quantities by the RG scale as was done for the Burgers equation in § 5.2. The two coefficients $D_\kappa$ and $\nu _\kappa$ are associated with anomalous dimensions $\eta _\kappa ^{D} = -\partial _s \ln D_\kappa$ and $\eta ^{\nu }_\kappa = -\partial _s \ln \nu _\kappa$. As one expects a power-law behaviour for the coefficient $\nu _\kappa$ beyond a certain scale, e.g. the Kolmogorov scale $\eta ^{-1}$, one can relate it to the physical viscosity $\nu \equiv \nu _\varLambda$ as

(6.9)\begin{equation} \nu_\kappa = \nu_{\eta^{{-}1}} ({\kappa}\eta)^{-\eta^{\nu}} \simeq \nu_{\varLambda} ({\kappa}\eta)^{-\eta^{\nu}} . \end{equation}

Because of Galilean invariance, the two anomalous dimensions $\eta _\kappa ^{D}$ and $\eta _\kappa ^{\nu }$ are not independent as in the Burgers case and satisfy the same exact relation (5.12), which can be used to eliminate for instance $\eta _\kappa ^{\nu }$ as

(6.10)\begin{equation} \eta_\kappa^{\nu} = (4-d+\eta_\kappa^{D})/3 . \end{equation}

The running anomalous dimension $\eta ^{D}_\kappa$ should be determined by computing the flow equation for $D_\kappa$, which has to be integrated along with the flow equations for $f_{\kappa,\perp }^{\nu }(\boldsymbol {p})$ and $f_{\kappa,\perp }^{\nu }(\boldsymbol {p})$. In the case of fully developed turbulence, the value of $\eta ^{D}_\kappa$ can be inferred by requiring a stationary state, that is that the mean injection rate balances the mean dissipation rate all along the flow. The average injected power by unit mass at the scale $\kappa$ can be expressed as (Canet et al. Reference Canet, Delamotte and Wschebor2016)

(6.11)\begin{align} \bar\epsilon =\langle \epsilon_{inj} \rangle &= \left\langle f_\alpha(t,\boldsymbol{x}) {{v}}_\alpha(t,\boldsymbol{x}) \right\rangle =\lim_{\delta t\to 0^{+}}\int_{\boldsymbol{x}'} N_{\kappa,\alpha\beta}(|\boldsymbol{x}-\boldsymbol{x}'|)\, G^{u\bar u}_{\kappa,\alpha\beta}(t+\delta t,\boldsymbol{x};t,\boldsymbol{x}')\nonumber\\ &=D_\kappa \kappa^{d} \lim_{\delta t\to 0^{+}} \left[(d-1) \int_{\hat \omega,\hat{\boldsymbol{q}}} \hat n(\hat{\boldsymbol{q}}) \exp({-{\rm i}\hat \omega \hat {\delta t}})\hat G^{u\bar u}_{\kappa,\perp}(\hat \omega,\hat{\boldsymbol{q}}) \right]\nonumber\\ & \equiv D_\kappa \kappa^{d} \hat{\gamma}, \end{align}

where $\hat {\gamma }$ depends only on the forcing profile since the frequency integral of the response function is one because of causality. It is given by

(6.12)\begin{equation} \hat{\gamma} = (d-1)\int\frac{d^{d} \hat{\boldsymbol{q}}}{(2{\rm \pi})^{d}} \hat n(\hat{\boldsymbol{q}}) = (d-1)\frac{2 {\rm \pi}^{d/2}}{(2{\rm \pi})^{d}\varGamma(d/2)}\int_0^{\infty} \,{\rm d}\hat{q}\, \hat{q}^{d-1}\hat n(\hat q), \end{equation}

where the last identity holds for an isotropic forcing.

The conditions for stationarity depend on the dimension, and in particular on the main form of energy dissipation, which differs in $d=3$ and $d=2$. Let us focus in the following on $d=3$. One can show that in this dimension, the dissipation at the Kolmogorov scale prevails on the dissipation at the boundary mediated by the regulator $R_\kappa$, and that it does not depend on $\kappa$ (it is given through an integral dominated by its UV bound $\propto \eta ^{-1}$). One concludes that to obtain a steady state, the mean injection should balance the mean dissipation and thus it should not depend on $\kappa$ either. This requires $D_\kappa \sim \kappa ^{-d}$, that is $\eta _\kappa ^{D} = d =3$. The value of $\eta ^{\nu }$ is then fixed by (6.10) to $\eta ^{\nu }=4/3$.

Thus, one may express the running coefficient $\nu _\kappa$ using (6.9) and the coefficient $D_\kappa$ in term of the injection rate using (6.11) as

(6.13a,b)\begin{equation} D_\kappa = \frac{\bar\epsilon}{ \hat\gamma} \kappa^{{-}3}, \quad \nu_\kappa = \bar\epsilon^{1/3}\kappa^{{-}4/3} , \end{equation}

where we used the definition of the Kolmogorov scale as $\nu _\varLambda =\nu =\bar \epsilon ^{1/3}\eta ^{4/3}$.

Finally, the scaling dimension of any correlation or vertex function can be deduced from the scaling dimensions of the fields (as determined in § 5.2). This yields for instance for the two-point correlation function

(6.14)\begin{equation} \bar G^{uu}_{\kappa,\perp}(\omega,\boldsymbol{p}) = \frac{\bar\epsilon^{1/3}}{\hat{\gamma}\kappa^{13/3}} \hat{G}^{uu}_{\kappa,\perp}\left(\hat{\omega} = \frac{\omega}{\bar\epsilon^{1/3} \kappa^{2/3}},\hat{\boldsymbol{p}}=\frac{\boldsymbol{p}}{\kappa}\right) . \end{equation}

6.3. Fixed-point renormalisation functions

Using the analysis of § 6.2, we define the dimensionless functions $\hat h_\kappa ^{\nu }$ and $\hat h_\kappa ^{D}$ as

(6.15a,b)\begin{equation} f_{\kappa,\perp}^{\nu}(\boldsymbol{p}) = \nu_\kappa \,\kappa^{2}\,\hat p^{2}\, \hat h_\kappa^{\nu}(\hat{\boldsymbol{p}}) \quad{\rm and} \quad f_{\kappa,\perp}^{D}(\boldsymbol{p}) = D_\kappa \, \hat p^{2}\, \hat h_\kappa^{D}(\hat{\boldsymbol{p}}) . \end{equation}

The flow equations of $\hat h_\kappa ^{\nu }$ and $\hat h_\kappa ^{D}$ are given by

(6.16)\begin{equation} \left. \begin{aligned} \partial_s \hat h_\kappa^{\nu}(\hat{\boldsymbol{p}}) & = \eta_\kappa^{\nu} \hat h_\kappa^{\nu}(\hat{\boldsymbol{p}}) +\hat{\boldsymbol{p}} \partial_{\hat{\boldsymbol{p}}} \hat h_\kappa^{\nu} (\hat{\boldsymbol{p}}) + \frac{\partial_s f_{\kappa,\perp}^{\nu}(\boldsymbol{p})}{\nu_\kappa \kappa^{2} \hat p^{2}}, \\ \partial_s \hat h_\kappa^{D}(\hat{\boldsymbol{p}}) & = (\eta_\kappa^{D}+2) \hat h_\kappa^{D}(\hat{\boldsymbol{p}}) +\hat{\boldsymbol{p}} \partial_{\hat{\boldsymbol{p}}} \hat h_\kappa^{D} (\hat{\boldsymbol{p}}) + \frac{\partial_s f_{\kappa,\perp}^{D}(\boldsymbol{p})}{D_\kappa\hat p^{2}} \end{aligned} \right\} \end{equation}

with the substitutions for dimensionless quantities in the flow equations (B8) and (B9) for $\partial _s f_{\kappa,\perp }^{\nu }(\boldsymbol {p})$ and $\partial _s f_{\kappa,\perp }^{D}(\boldsymbol {p})$.

These flow equations are first-order differential equations in the RG scale $\kappa$, they can be integrated numerically from the initial condition (6.6a,b), which corresponds to $\hat h_\kappa ^{\nu }(\hat {\boldsymbol {p}})=1$ and $\hat h_\kappa ^{D}(\hat {\boldsymbol {p}})=0$, down to $\kappa \to 0$. For this, the wavevectors are discretised on a (modulus, angle) grid. At each RG time step $s$, the derivatives are computed using five-point finite differences, the integrals are computed using Gauss–Legendre quadrature, with both interpolation and extrapolation procedures to evaluate combinations $\boldsymbol {p}+\boldsymbol {q}$ outside of the mesh points. One observes that the functions $\hat h_\kappa ^{\nu }$ and $\hat h_\kappa ^{D}$ smoothly deform from their constant initial condition to reach a fixed point where they stop evolving after a typical RG time $s\lesssim -10$. This is illustrated on figure 4, where the fixed-point functions recorded at $s=-25$ are highlighted with a thick line. This result shows that the fully developed turbulent state corresponds to a fixed point of the RG flow, which means that it is scale invariant. However, this fixed point exhibits a very peculiar feature.

Figure 4. The RG evolution of the renormalisation functions $\hat h_\kappa ^{\nu }$ and $\hat h_\kappa ^{D}$ with the RG scale, from constant initial conditions (black horizontal lines) to their fixed-point shape (bold blue lines). The red arrows indicate the RG flow, decreasing the RG time from $s=0$ to $s=-25$. Insets: local exponents, defined by (6.18a,b), at the fixed point, with the corresponding K41 values indicated as dashed lines.

Indeed, the fixed-point functions are found to behave as power laws at large wavenumbers as expected. However, the corresponding exponents differ from their K41 values. The fixed-point functions can be described at large $\hat {p}$ as

(6.17a,b)\begin{equation} \hat h_*^{\nu}(\hat p) \sim \hat p^{-\eta^{\nu}+\delta \eta^{\nu}} \quad\text{and} \quad \hat h_*^{D}(\hat p) \sim \hat p^{-(\eta^{D}+2)+\delta \eta^{D}}, \end{equation}

where $\delta \eta ^{\nu }$ and $\delta \eta ^{D}$ are the deviations from K41 scaling. The insets of figure 4 show the actual local exponents $\eta ^{D}_{loc}$ and $\eta ^{\nu }_{loc}$ at the fixed point defined as

(6.18a,b)\begin{equation} \eta^{D}_{loc} = \frac{{\rm d}\ln \hat h_*^{D}(\hat p)}{{\rm d}\ln\hat p}\quad\eta^{\nu}_{loc} = \frac{{\rm d}\ln \hat h_*^{\nu}(\hat p)}{{\rm d}\ln\hat p}. \end{equation}

When a function $f(x)$ behaves as a power law $f(x)\sim x^{\alpha }$ in some range, the local exponent defined by this logarithmic derivative identifies with $\alpha$ on this range. It is clear that $\eta ^{\nu }_{loc}$ differs from its expected K41 value $\eta ^{\nu }=4/3$, and similarly for $\eta ^{D}_{loc}$. The deviations are estimated numerically as $\delta \eta ^{\nu }\simeq \delta \eta ^{D}\simeq 0.33$.

Let us emphasise that it is a very unusual feature, which originates in the breaking of the decoupling property in the NS flow equations. Non-decoupling means that the loop contributions ${\partial _s f_{\kappa,\perp }^{D}(\boldsymbol {p})}/{D_\kappa \hat p^{2}}$ and ${\partial _s f_{\kappa,\perp }^{\nu }(\boldsymbol {p})}/{\nu _\kappa \kappa ^{2} \hat p^{2}}$ in the flow equations (6.16) do not become negligible in the limit of large wavenumbers compared with the linear terms. Intuitively, a large wavenumber is equivalent to a large mass, and degrees of freedom with a large mass are damped and do not contribute in the dynamics at large (non-microscopic) scale. This means that the IR (effective) properties are not affected by the UV (microscopic) details, and this pertains to the mechanism for universality. For turbulence, this is not the case, and the consequences of this breaking, intimately related to sweeping, will be further expounded in § 7. Within the simple LO approximation, the signature of this non-decoupling is that the exponent of the power laws at large $p$ are not fixed by the scaling dimensions $\eta _*^{\nu }$ and $\eta _*^{D}$, or in other words, the scaling behaviours in $\kappa$ and in $\boldsymbol {p}$ are different. Indeed, the scaling in $\kappa$ is fixed by (6.10), but $\eta _\kappa ^{D}$ and $\eta _\kappa ^{\nu }$ do not control the scaling behaviour in $\boldsymbol {p}$, given by (6.17a,b) which exhibit explicit deviations $\delta \eta ^{\nu }$ and $\delta \eta ^{D}$. This is in sharp contrast with what was observed for the Burgers–KPZ case in § 5, where decoupling was satisfied, and the behaviour in $\boldsymbol {p}$ and $\omega$, e.g. in the scaling form (5.16) is indeed controlled by $\eta _\kappa$. However, it turns out that this peculiar feature plays no role for equal-time quantities. As we show in the next section, the LO approximation leads to Kolmogorov scaling for the energy spectrum and the structure functions. Hence the deviations $\delta \eta ^{\nu }$ and $\delta \eta ^{D}$ are a priori not directly observable.

6.4. Kinetic energy spectrum

Let us now compute from this fixed point physical observables, starting with the kinetic energy spectrum. The mean total energy per unit mass is given by

(6.19)\begin{equation} \frac 1 2 \langle \boldsymbol{v}(t,\boldsymbol{x})^{2}\rangle = \frac 1 2 G_{\alpha\alpha}^{uu}(0,0) = \frac 1 2 \int \frac{d^{d} \boldsymbol{p}}{(2{\rm \pi})^{d}} \int \frac{{\rm d}\omega}{2{\rm \pi}} \bar G_{\alpha\alpha}^{uu}(\omega,\boldsymbol{p}) . \end{equation}

The kinetic energy spectrum, defined as the energy density at wavenumber $p$, is hence given for an isotropic flow by

(6.20)\begin{equation} {\mathcal{E}}(p) = \frac 1 2 \frac{2{\rm \pi}^{d/2}}{(2{\rm \pi})^{d} \varGamma(d/2)} p^{d-1} (d-1) \int \frac{{\rm d}\omega}{2{\rm \pi}} \bar G_{{\perp}}^{uu}(\omega,p). \end{equation}

The statistical properties of the system are obtained once all fluctuations have been integrated over, i.e. in the limit $\kappa \to 0$. Since the RG flow reaches a fixed point, the limit $\kappa \to 0$ simply amounts to evaluating at the fixed point. Within the LO approximation, and in $d=3$, one finds

(6.21)\begin{equation} {\mathcal{E}}(p) = \displaystyle\frac{\bar\epsilon^{2/3}}{\hat \gamma \kappa^{5/3}} \hat{E}(\hat p) ,\quad\text{with}\ \hat{E}(\hat p) = \frac{1}{2{\rm \pi}^{2}}\, \hat p^{2}\,\frac{\hat h_*^{D}(\hat{p})}{\hat h_*^{\nu}(\hat p)}. \end{equation}

The function $\hat {E}(\hat p)$ is represented in figure 5, with in the inset the compensated spectrum $\hat {p}^{5/3}\hat {E}(\hat p)$. At small $\hat p$, it behaves as $\hat p^{2}$, which reflects equipartition of energy. At large $\hat p$, the two corrections $\delta \eta ^{\nu }$ and $\delta \eta ^{D}$ turn out to compensate in (6.21), such that one accurately recovers the Kolmogorov scaling $\hat {E}(\hat p) \simeq a \hat p^{-5/3}$, with $a\simeq 0.106$ a numerical factor, which can be read off from the plateau value of the compensated spectrum $\hat {p}^{5/3}\hat {E}(\hat p)$ shown in the inset of figure 5. Thus one obtains for the spectrum

(6.22)\begin{equation} {\mathcal{E}}(p) = \frac{a}{\hat\gamma} \bar\epsilon^{2/3} p^{{-}5/3} \equiv C_K \bar\epsilon^{2/3}p^{{-}5/3} . \end{equation}

For the present choice of forcing profile, which is $\hat n(\hat x) = \hat x^{2} \textrm {e}^{-\hat x^{2}}$, one obtains from (6.12) $\hat \gamma = \tfrac {3}{8}{\rm \pi} ^{-3/2}$, and thus $C_K\simeq 1.572$. This value is in precise agreement with typical experimental values, as compiled e.g. by Sreenivasan (Reference Sreenivasan1995), which yields $C_K\simeq 1.52\pm 0.30$. Note that perturbative RG approaches yielded the estimate $C_K=1.617$ (Yakhot & Orszag Reference Yakhot and Orszag1986; Dannevik, Yakhot & Orszag Reference Dannevik, Yakhot and Orszag1987) although from an uncontrolled limit. Kraichnan had to resort to an improved DIA (direct interaction approximation) scheme, called Lagrangian history DIA, to obtain the correct K41 spectrum and an estimate of this constant $C_K=1.617$ (Kraichnan Reference Kraichnan1965).

Figure 5. (a) Kinetic energy spectrum (main plot) and compensated one (inset); (b) second-order structure function (main plot) and compensated one (inset); (c) third-order structure function (main plot) and compensated one (inset). All functions are calculated from the FRG fixed point obtained within the LO approximation in $d=3$.

Let us emphasise that $C_K$ explicitly depends on the forcing profile through $\hat \gamma$. It is thus a priori non-universal. However, it only depends on the integral of this profile, which is required to vanish at zero, be peaked around the integral scale and decay fast after. Thus one can expect that this integral is not very sensitive to the precise shape of the forcing satisfying these constraints. Indeed, it was shown in Tomassini (Reference Tomassini1997) within an approximation similar to LO that the numerical value of $C_K$