Anomalous scaling of a passive scalar advected by the Navier--Stokes velocity field: Two-loop approximation

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non-Gaussian velocity field with finite correlation time. The velocity is governed by the Navier--Stokes equation, subject to an external random stirring force with the correlation function $\propto \delta(t-t') k^{4-d-2\epsilon}$. It is shown that the scalar field is intermittent already for small $\epsilon$, its structure functions display anomalous scaling behavior, and the corresponding exponents can be systematically calculated as series in $\epsilon$. The practical calculation is accomplished to order $\epsilon^{2}$ (two-loop approximation), including anisotropic sectors. Like for the well-known Kraichnan's rapid-change model, the anomalous scaling results from the existence in the model of composite fields (operators) with negative scaling dimensions, identified with the anomalous exponents. Thus the mechanism of the origin of anomalous scaling appears similar for the Gaussian model with zero correlation time and non-Gaussian model with finite correlation time. It should be emphasized that, in contrast to Gaussian velocity ensembles with finite correlation time, the model and the perturbation theory discussed here are manifestly Galilean covariant. The relevance of these results for the real passive advection, comparison with the Gaussian models and experiments are briefly discussed.Comment: 25 pages, 1 figur

Re-examination of the infra-red properties of randomly stirred hydrodynamics

Dynamic renormalization group (RG) methods were originally used by Forster, Nelson and Stephen (FNS) to study the large-scale behaviour of randomly-stirred, incompressible fluids governed by the Navier-Stokes equations. Similar calculations using a variety of methods have been performed since, but have led to a discrepancy in results. In this paper, we carefully re-examine in $d$-dimensions the approaches used to calculate the renormalized viscosity increment and, by including an additional constraint which is neglected in many procedures, conclude that the original result of FNS is correct. By explicitly using step functions to control the domain of integration, we calculate a non-zero correction caused by boundary terms which cannot be ignored. We then go on to analyze how the noise renormalization, absent in many approaches, contributes an ${\mathcal O}(k^2)$ correction to the force autocorrelation and show conditions for this to be taken as a renormalization of the noise coefficient. Following this, we discuss the applicability of this RG procedure to the calculation of the inertial range properties of fluid turbulence.Comment: 16 pages, 6 figure

Stability of scaling regimes in d≥2d\geq 2 developed turbulence with weak anisotropy

The fully developed turbulence with weak anisotropy is investigated by means of renormalization group approach (RG) and double expansion regularization for dimensions $d\ge 2$. Some modification of the standard minimal substraction scheme has been used to analyze stability of the Kolmogorov scaling regime which is governed by the renormalization group fixed point. This fixed point is unstable at $d=2$; thus, the infinitesimally weak anisotropy destroyes above scaling regime in two-dimensional space. The restoration of the stability of this fixed point, under transition from $d=2$ to $d=3,$ has been demonstrated at borderline dimension $2<d_c<3$. The results are in qualitative agreement with ones obtained recently in the framework of the usual analytical regularization scheme.Comment: 23 pages, 2 figure

Renormalization group in the infinite-dimensional turbulence: third-order results

The field theoretic renormalization group is applied to the stochastic Navier-Stokes equation with the stirring force correlator of the form k^(4-d-2\epsilon) in the d-dimensional space, in connection with the problem of construction of the 1/d expansion for the fully developed fluid turbulence beyond the scope of the standard epsilon expansion. It is shown that in the large-d limit the number of the Feynman diagrams for the Green function (linear response function) decreases drastically, and the technique of their analytical calculation is developed. The main ingredients of the renormalization group approach -- the renormalization constant, beta function and the ultraviolet correction exponent omega, are calculated to order epsilon^3 (three-loop approximation). The two-point velocity-velocity correlation function, the Kolmogorov constant C_K in the spectrum of turbulent energy and the inertial-range skewness factor S are calculated in the large-d limit to third order of the epsilon expansion. Surprisingly enough, our results for C_K are in a reasonable agreement with the existing experimental estimates.Comment: 30 pages with EPS figure

Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow

Field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the Gaussian non-solenoidal (``compressible'') velocity field with the covariance $\propto\delta(t-t')| x-x'|^{\epsilon}$. Convective range anomalous scaling for the structure functions and various pair correlators is established, and the corresponding anomalous exponents are calculated to the order $\epsilon^2$ of the $\epsilon$ expansion. These exponents are non-universal, as a result of the degeneracy of the RG fixed point. In contrast to the case of a purely solenoidal velocity field (Obukhov--Kraichnan model), the correlation functions in the case at hand exhibit nontrivial dependence on both the IR and UV characteristic scales, and the anomalous scaling appears already at the level of the pair correlator. The powers of the scalar field without derivatives, whose critical dimensions determine the anomalous exponents, exhibit multifractal behaviour. The exact solution for the pair correlator is obtained; it is in agreement with the result obtained within the $\epsilon$ expansion. The anomalous exponents for passively advected magnetic fields are also presented in the first order of the $\epsilon$ expansion.Comment: 31 pages, REVTEX file. More detailed discussion of the one-dimensional case and comparison to the previous paper [20] are given; references updated. Results and formulas unchange

Anomalous scaling of passively advected magnetic field in the presence of strong anisotropy

Inertial-range scaling behavior of high-order (up to order N=51) structure functions of a passively advected vector field has been analyzed in the framework of the rapid-change model with strong small-scale anisotropy with the aid of the renormalization group and the operator-product expansion. It has been shown that in inertial range the leading terms of the structure functions are coordinate independent, but powerlike corrections appear with the same anomalous scaling exponents as for the passively advected scalar field. These exponents depend on anisotropy parameters in such a way that a specific hierarchy related to the degree of anisotropy is observed. Deviations from power-law behavior like oscillations or logarithmic behavior in the corrections to structure functions have not been found.Comment: 15 pages, 18 figure

Fluctuating hydrodynamics and turbulence in a rotating fluid: Universal properties

We analyze the statistical properties of three-dimensional ($3d$) turbulence in a rotating fluid. To this end we introduce a generating functional to study the statistical properties of the velocity field $\bf v$. We obtain the master equation from the Navier-Stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity $\mathbf \Omega$. In particular we obtain the {\em differential forms} for the analogs of the well-known von Karman-Howarth relation for $3d$ fluid turbulence. We examine their behavior in the limit of large rotation. Our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to $\mathbf \Omega$. The hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two-dimensional flows. To complement these results, by using an effective model in the small-$\Omega$ limit, within a one-loop approximation, we show that the equal-time correlation of the velocity components parallel to $\mathbf \Omega$ displays Kolmogorov scaling $q^{-5/3}$, where as for all other components, the equal-time correlators scale as $q^{-3}$ in the inertial range where $\bf q$ is a wavevector in $3d$. Our results are generally testable in experiments and/or direct numerical simulations of the Navier-Stokes equation in a rotating frame.Comment: 24 pages in preprint format; accepted for publication in Phys. Rev. E (2011

Study of Anomalous Kinetics of The Annihilation Reaction A+A->0

Using the perturbative renormalization group, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction A+A->0 at and below its critical dimension d_c=2. We use the second-quantization formalism of Doi to bring the stochastic problem to a field-theoretic form. We investigate the reaction in the vicinity of the space dimension d=2 using a two-parameter expansion in $\epsilon$ and $\Delta$, where $\epsilon$ is the deviation from the Kolmogorov scaling parameter and $\Delta$ is the deviation from the space dimension d=2. We evaluate all the necessary quantities, including fixed points with their regions of stability, up to the second order of the perturbation theory.Comment: Presented in the Third International Conference "Models in QFT: In Memory of A.N. Vasiliev" St. Petersburg - Petrodvorez, October 201

A non-perturbative renormalization group study of the stochastic Navier--Stokes equation

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 non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the H\"older exponent $4-2\,\varepsilon$ 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 $\varepsilon=2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.Comment: RevTeX, 18 pages, 5 figures. Minor changes and new discussion