An improved \eps expansion for three-dimensional turbulence: summation of nearest dimensional singularities

An improved \eps expansion in the $d$-dimensional ($d > 2$) stochastic theory of turbulence is constructed by taking into account pole singularities at $d \to 2$ in coefficients of the \eps expansion of universal quantities. Effectiveness of the method is illustrated by a two-loop calculation of the Kolmogorov constant in three dimensions.Comment: 4 page

Anomalous scaling, nonlocality and anisotropy in a model of the passively advected vector field

A model of the passive vector quantity advected by a Gaussian time-decorrelated self-similar velocity field is studied; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by an infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions. The exponents are organized in a hierarchical order according to their degree of anisotropy, with the spectrum unbounded from above and the leading exponent coming from the isotropic sector. For the higher-order structure functions, the anomalous scaling behavior is a consequence of the existence in the corresponding operator product expansions of ``dangerous'' composite operators, whose negative critical dimensions determine the exponents. A close formal resemblance of the model with the stirred NS equation reveals itself in the mixing of operators. Using the RG, the anomalous exponents are calculated in the one-loop approximation for the even structure functions up to the twelfth order.Comment: 37 pages, 4 figures, REVTe

Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ε3\varepsilon^{3}

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function \propto\delta(t-t')/k^{d+\eps}. Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the $\varepsilon$ expansion to order $\varepsilon^{3}$ (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the $\varepsilon$ expansion are discussed; the improved ``inverse'' $\varepsilon$ expansion is proposed and the comparison with the existing nonperturbative results is given.Comment: 34 pages, 5 figures, REVTe

Stochastic magnetohydrodynamic turbulence in space dimensions d≥2d\ge 2

Interplay of kinematic and magnetic forcing in a model of a conducting fluid with randomly driven magnetohydrodynamic equations has been studied in space dimensions $d\ge 2$ by means of the renormalization group. A perturbative expansion scheme, parameters of which are the deviation of the spatial dimension from two and the deviation of the exponent of the powerlike correlation function of random forcing from its critical value, has been used in one-loop approximation. Additional divergences have been taken into account which arise at two dimensions and have been inconsistently treated in earlier investigations of the model. It is shown that in spite of the additional divergences the kinetic fixed point associated with the Kolmogorov scaling regime remains stable for all space dimensions $d\ge 2$ for rapidly enough falling off correlations of the magnetic forcing. A scaling regime driven by thermal fluctuations of the velocity field has been identified and analyzed. The absence of a scaling regime near two dimensions driven by the fluctuations of the magnetic field has been confirmed. A new renormalization scheme has been put forward and numerically investigated to interpolate between the $\epsilon$ expansion and the double expansion.Comment: 12 pages, 4 figure

Anomalous scaling of a passive scalar in the presence of strong anisotropy

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar field, advected by the Gaussian strongly anisotropic velocity field. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the n-th order structure functions of scalar field are obtained; they are represented by superpositions of power laws with nonuniversal (dependent on the anisotropy parameters) anomalous exponents. In the limit of vanishing anisotropy, the exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. For the finite anisotropy, the exponents cannot be associated with individual operators (which are essentially ``mixed'' in renormalization), but the aforementioned hierarchy survives for all the cases studied. The second-order structure function is studied in more detail using the renormalization group and zero-mode techniques.Comment: REVTEX file with EPS figure

Anomalous exponents in the rapid-change model of the passive scalar advection in the order ϵ3\epsilon^{3}

Field theoretic renormalization group is applied to the Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance $- <{\bf v}(t,{\bf x}){\bf v}(t',{\bf x'})> \propto\delta(t-t')|{\bf x}-{\bf x'} |^{\epsilon}$. Inertial-range anomalous exponents, related to the scaling dimensions of tensor composite operators built of the scalar gradients, are calculated to the order $\epsilon^{3}$ of the $\epsilon$ expansion. The nature and the convergence of the $\epsilon$ expansion in the models of turbulence is are briefly discussed.Comment: 4 pages; REVTeX source with 3 postscript figure

Theory of the nonsteady diffusion growth of a gas bubble in a supersaturated solution of gas in liquid

Using a self-similar approach a general nonsteady theory is elaborated for the case of the diffusion growth of a gas bubble in a supersaturated solution of gas in liquid. Due to the fact that the solution and the bubble in it are physically isolated, the self-similar approach accounts for the balance of the number of gas molecules in the solution and in the bubble that expells incompressible liquid solvent while growing. The rate of growth of the bubble radius in its dependence from gas solubility and solution supersaturation is obtained. There is a nonsteady effect of rapid increase of the rate of bubble growth simultaneous with the growth of the product of gas solubility and solution supersaturation. This product is supplied with a limitation from above, which also stipulates isothermal conditions of bubble growth. The smallness of gas solubility is not presupposed.Comment: 22 pages, 3 figure

Passive scalar turbulence in high dimensions

Exploiting a Lagrangian strategy we present a numerical study for both perturbative and nonperturbative regions of the Kraichnan advection model. The major result is the numerical assessment of the first-order $1/d$-expansion by M. Chertkov, G. Falkovich, I. Kolokolov and V. Lebedev ({\it Phys. Rev. E}, {\bf 52}, 4924 (1995)) for the fourth-order scalar structure function in the limit of high dimensions $d$'s. %Two values of the velocity scaling exponent $\xi$ have been considered: %$\xi=0.8$ and $\xi=0.6$. In the first case, the perturbative regime %takes place at $d\sim 30$, while in the second at $d\sim 25$, %in agreement with the fact that the relevant small parameter %of the theory is $\propto 1/(d (2-\xi))$. In addition to the perturbative results, the behavior of the anomaly for the sixth-order structure functions {\it vs} the velocity scaling exponent, $\xi$, is investigated and the resulting behavior discussed.Comment: 4 pages, Latex, 4 figure

Field theoretic renormalization group for a nonlinear diffusion equation

The paper is an attempt to relate two vast areas of the applicability of the renormalization group (RG): field theoretic models and partial differential equations. It is shown that the Green function of a nonlinear diffusion equation can be viewed as a correlation function in a field-theoretic model with an ultralocal term, concentrated at a spacetime point. This field theory is shown to be multiplicatively renormalizable, so that the RG equations can be derived in a standard fashion, and the RG functions (the $\beta$ function and anomalous dimensions) can be calculated within a controlled approximation. A direct calculation carried out in the two-loop approximation for the nonlinearity of the form $\phi^{\alpha}$, where $\alpha>1$ is not necessarily integer, confirms the validity and self-consistency of the approach. The explicit self-similar solution is obtained for the infrared asymptotic region, with exactly known exponents; its range of validity and relationship to previous treatments are briefly discussed.Comment: 8 pages, 2 figures, RevTe

Theory of Developed Turbulence: Principle of Maximal Randomness and Spontaneous Parity Violation

A set of self-consistent equations in one-loop approximation in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the principle of maximal randomness of the velocity field with a given energy spectral flux, is obtained. These equations do not possess both infrared and ultraviolet divergences near the Kolmogorov values of indices. The formal solution found of the system of equations yields Kolmogorov exponents, but this solution leads to negative Kolmogorov constant $C_{\rm k}$ and negative viscosity. It has been shown, that the turbulent fluid becomes stable if spontaneous parity violation is achieved. Namely, the solution with the Kolmogorov indices and additional helical terms (which lead to positive both $C_{\rm k}$ and effective viscosity) exists. This solution predicts a large, closed to limit value, helical coefficient $\Theta$ in inertial range. The relationship obtained between $C_{\rm k}$ and $\Theta$ confirms this conclusion for the experimental value of $C_{\rm k}$