Nonperturbative renormalization group study of the stochastic Navier-Stokes equation

Abstract

We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4 − 2 ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's −5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the −5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.

Functional renormalization group approach to the Kraichnan model

“…In the FRG framework one considers a scale dependent effective action, the Effective Average Action (EAA), which interpolates between the classical action and the standard effective action [14][15][16]. Within this framework the issue of a fixed point for the functional integral associated to the randomly stirred Navier-Stokes equation has been first considered in [17] and more recently in [18] and [19,20]. In particular in [20] a fixed * capagani@uni-mainz.de point solution of the FRG equation associated to this problem has been investigated in d = 2 and d = 3.…”

Spatio-temporal correlations in 3D homogeneous isotropic turbulence

“…An interpretation of this result can be proposed based on the identification of the advection-velocity correlation function T at t = 0 as the spectral energy transfer function. We have observed that a significant part of the energy transfers between the small scales in 3D turbulence occurs in spectral triads with participation of a large scale mode as mediator (the term T SLS in the decomposition (28)). The same conclusion can be found in Ref.…”

“…The FRG is a versatile method well-developed since the early 1990's and used in a wide range of applications, both in high-energy physics (quantum gravity and QCD), condensed matter, quantum many-particle systems and statistical mechanics, including disordered and nonequilibrium problems (see References 23-26 for reviews). This method has been employed in particular to study the incompressible 3D Navier-Stokes equation in several works 14,[27][28][29][30][31] . We here focus on a recent result concerning the spatio-temporal dependence of multi-point correlation functions of the turbulent velocity field in homogeneous, isotropic and stationary conditions.…”

Anomalous scaling at nonthermal fixed points of Burgers' and Gross-Pitaevskii turbulence

“…According to the definition (5) the above scaling form is consistent with the relations (29), (34)-(36) between χ, z, η 1 , and d. Moreover, the factor k η1−d /z 1 , which makes the argument of g dimensionless, does not depend on the cutoff scale k, see Eq. (26), and is used to normalise the scaling function. See the discussion in Sect.…”