Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models

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 the limiting case of vanishing correlation time. This 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 $\mathcal{E}\propto k^{1-y}$ and the dispersion law $\omega \propto k^{2-\eta }$. 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.

Figure 1

Graphical representation of the bare propagators in the model (3.1).

Figure 2

Graphical representation of the interaction vertex in the model (3.1).

Figure 3

The one-loop approximation of the 1-irreducible response function ${\langle v_{\alpha }^{\prime }{v}_{\beta }\rangle }_{1-\mathrm{ir}}$. Greek letters denote external (free) indices of the diagram; Latin letters correspond to internal indices of the vector fields with implied summation.

Figure 4

The one-loop approximation of the 1-irreducible function ${\langle v_{\alpha }^{\prime }{v}_{\beta }{v}_{\gamma }\rangle }_{1-\mathrm{ir}}$. Greek letters denote external (free) indices of the diagram; Latin letters correspond to internal indices of the vector fields with implied summation.

Figure 5

Domains of the existence and IR stability of the fixed points for the model (3.1) in the plane $(y,\eta )$; $a^{*}=\frac{4/3}{3d(d-1)+2}$. Gray areas correspond to the regions where the trivial and $\delta$ correlated model fixed points are IR attractive; hatched areas which are lying inside the gray ones correspond to the regions where another nontrivial (saddle type) point exists; in white areas there is no IR attractive fixed point.

Figure 6

Graphical representation of the bare propagators ${\langle \theta {\theta }^{\prime }\rangle }_0$ and ${\langle \theta \theta \rangle }_0$.

Figure 7

Graphical representation of the interaction vertex $V_j$.

Figure 8

The one-loop approximation of the function ${\mathrm{\Gamma }}_n(x;\theta )$. Latin letters $i$ and $j$ denote the internal indices of the velocity field with implied summation.

Figure 9

The graph of the function ${\mathcal{S}}_l(3/2)$ as a function of $l$.