Quantum-field renormalization group in the theory of developed turbulence: Consideration of anisotropy and passive scalar admixtures

The d-dimensional model of fully developed homogeneous anisotropic turbulence is studied by the quantum-field renormalization group method. Generalization of the model to systems with a passive scalar admixture is also considered. It is shown that in the case of weak anisotropy, the fixed point of the isotropic model remains stable for d>2.68 and Kolmogrov's scaling law still holds. Corrections to the Prandtl number are found.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 412–422, December, 1995.     

Transfer of a Passive Vector Admixture by a Two-Dimensional Turbulent Flow

We consider a model of a passive vector field transfer by a random two-dimensional transverse velocity field that is uncorrelated in time and has Gaussian spatial statistics given by a powerlike correlator. We use the renormalization group and the operator product expansion techniques to show that the asymptotic approximation of the structure functions of a vector field in the inertial range is determined by the energy dissipation fluctuations. The dependence of the asymptotic approximation on the external scale of turbulence is essential and has a powerlike form (the case of an anomalous scaling). The corresponding exponents are calculated in the one-loop approximation for structure functions of an arbitrary order.     

Renormalization group in the problem of fully developed turbulence of a compressible fluid

A model of fully developed turbulence of a compressible liquid (gas), based on the stochastic Navier-Stokes equation, is considered by means of the renormalization group. It is proved that the model is multiplicatively renormalized in terms of the “velocity-logarithm of density” variables. The scaling dimensions of the fields and parameters are calculated in the one-loop approximation. Dependence of the effective sound velocity and the Mach number on the integral turbulence scale L is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 385–398, March, 1997.     

Anomalous scaling in the model of turbulent advection of a vector field

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.     

Renormalization group approach and short-distance expansion in theory of developed turbulence: Asymptotics of the triplex equal-time correlation function

Asymptotics of the triplex equal-time correlation function for the turbulence developed in incompressible fluids in the region of widely separated wave vector values is investigated using the renormalization group approach and short-distance expansion. The problem of the most essential composite operators contributing to these asymptotics is examined. For this purpose, the critical dimensions of a family of composite quadratic tensor operators in the velocity gradient are found. Considered in the one-loop approximation, the contribution of these operators turns out to be less substantial (although not significantly) than the contribution of the linear term. The derived asymptotics of the triplex correlator coincide in form with that predicted by the EDQNM approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 450–461, December, 1995.     

Renormalization group in the theory of turbulence: Three-loop approximation as <Emphasis Type="Italic">d</Emphasis> → ∞

We use the renormalization group method to study the stochastic Navier-Stokes equation with a random force correlator of the form k ^4−d−2ɛ in a d-dimensional space in connection with the problem of constructing a 1/d-expansion and going beyond the framework of the standard ɛ-expansion in the theory of fully developed hydrodynamic turbulence. We find a sharp decrease in the number of diagrams of the perturbation theory for the Green’s function in the large-d limit and develop a technique for calculating the diagrams analytically. We calculate the basic ingredients of the renormalization group approach (renormalization constant, β-function, fixed-point coordinates, and ultraviolet correction index ω) up to the order ɛ ³ (three-loop approximation). We use the obtained results to propose hypothetical exact expressions (i.e., not in the form of ɛ-expansions) for the fixed-point coordinate and the index ω. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 460–477, March, 2009.     

An algorithm for studying the renormalization group dynamics in the projective space

The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.     

Anomalous scaling in statistical models of passively advected vector fields

We use the methods of the renormalization group and the operator product expansion to consider the problem of the stochastic advection of a passive vector field with the most general form of the nonlinear term allowed by the Galilean symmetry. The external velocity field satisfies the Navier-Stokes equation. We show that the correlation functions have anomalous scaling in the inertial range. The corresponding anomalous exponents are determined by the critical dimensions of tensor composite fields (operators) built from only the fields themselves. We calculate the anomalous dimensions in the leading order of the expansion in the exponent in the correlator of the external force in the Navier-Stokes equation (the oneloop approximation of the renormalization group). The anomalous exponents exhibit a hierarchy related to the anisotropy degree: the lower the rank of the tensor operator is, the lower its dimension. The leading asymptotic terms are determined by the scalar operators in both the isotropic and the anisotropic cases, which completely agrees with Kolmogorov’s hypothesis of local isotropy restoration.     

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.     

A one-dimensional model for dispersive wave turbulence

Summary A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|^−1/3) spectrum compared with the steeper (|k|^−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.     

Renormalization Group and the Ricci Flow

We discuss from a geometric point of view the connection between the renormalization group flow for non–linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow.     

Renormalization group in the theory of fully developed turbulence. Composite operators of canonical dimension 8

Renormalization and critical dimensions of the family of Galilean invariant scalar composite operators of canonical dimension eight are considered within the framework of the renormalization group approach to the stochastic theory of fully developed turbulence.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 92–101, January, 1996.     

Influence of Finite-Time Velocity Correlations on Scaling Properties of the Magnetic Field in the Kazantsev-Kraichnan Model: Two-Loop Renormalization Group Analysis

Theoretical and Mathematical Physics - Using the field theory renormalization group method and the operator product expansion technique in the two-loop approximation, we investigate the influence...     

Hexagonal Fourier-Galerkin Methods for the Two-Dimensional Homogeneous Isotropic Decaying Turbulence

In this paper, we propose two hexagonal Fourier-Galerkin methods for the direct numerical simulation of the two-dimensional homogeneous isotropic decaying turbulence. We first establish the lattice Fourier analysis as a mathematical foundation. Then a universal approximation scheme is devised for our hexagonal Fourier-Galerkin methods for Navier-Stokes equations. Numerical experiments mainly concentrate on the decaying properties and the self-similar spectra of the two-dimensional homogeneous turbulence at various initial Reynolds numbers with an initial flow field governed by a Gaussian-distributed energy spectrum. Numerical results demonstrate that both the hexagonal Fourier-Galerkin methods are as efficient as the classic square Fourier-Galerkin method, while provide more effective statistical physical quantities in general.     

Renormalization group in the theory of developed turbulence. The problem of justifying the Kolmogorov hypotheses for composite operators

In this paper, the stochastic theory of developed turbulence is considered within the framework of the quantum field renormalization group and operator expansions. The problem of justifying the Kolmogorov-Obukhov theorem in application to the correlation functions of composite operators is discussed. An explicit expression is found for the critical dimension of a general-type composite operator. For an arbitrary UV-finite composite operator, the second Kolmogorov hypothesis (the viscosity-independence of the correlator) is proved and the dependence of various correlators on the external turbulence scale is determined. It is shown that the problem involves an infinite number of Galilean-invariant scalar operators with negative critical dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 122–136, January, 1997.     

Renormalization group in a fermionic hierarchical model in projective coordinates

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.     

Renormalization-group approach to the problem of the effect of compressibility on the spectral properties of developed turbulence

Using the renormalization group technique, the spectra of the developed turbulence of a compressible liquid are investigated for the case of small Mach numbers. Composite operators are found that determine corrections to the spectra due to compressibility. Renormalization of these operators is studied and corresponding critical dimensions are obtained. The corrections are proved to be independent of viscosity in the inertial range as in the case of an incompressible liquid.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 260–270, August, 1995.