In this paper we consider the model of incompressible fluid described by the stochastic Navier-Stokes equation with finite correlation time of a random force. Inertial-range asymptotic behavior of fully developed turbulence is studied by means of the field theoretic renormalization group within the one-loop approximation. It is corroborated that regardless of the values of model parameters and initial data, the inertial-range behavior of the model is described by limiting case of vanishing correlation time. It indicates that the Galilean symmetry of the model violated by the "colored" random force is restored in the inertial range. This regime corresponds to the only nontrivial fixed point of the renormalization group equation. The stability of this point depends on the relation between the exponents in the energy spectrum ${\cal E} \propto k^{1-y}$ and the dispersion law $ω\propto k^{2-η}$. The second analyzed problem is the passive advection of a scalar field by this velocity ensemble. Correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. We demonstrate that in accordance with Kolmogorov's hypothesis of the local symmetry restoration, the main contribution to the operator product expansion is given by the isotropic operator, while anisotropic terms should be considered only as corrections.
Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model
On Functional and Holographic Renormalization Group Methods in Stochastic Theory of Turbulence
A nonlocal quantum-field model is constructed for the system of hydrodynamic equations for incompressible viscous fluid (the stochastic Navier--Stokes (NS) equation and the continuity equation). This model is studied by the following two mutually parallel methods: the Wilson--Polchinski functional renormalization group method (FRG), which is based on the exact functional equation for the generating functional of amputated connected Green's functions (ACGF), and the Heemskerk--Polchinski holographic renormalization group method (HRG), which is based on the functional Hamilton--Jacobi (HJ) equat…
