The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions

It is shown that the idea that scaling behavior in turbulence is limited by one outer length $L$ and one inner length $\eta$ is untenable. Every n'th order correlation function of velocity differences \bbox{\cal F}_n(\B.R_1,\B.R_2,\dots) exhibits its own cross-over length $\eta_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. One result of this Letter is that when all these separations are of the same order $R$ this length scales like $\eta_n(R)\sim \eta (R/L)^{x_n}$ with $x_n=(\zeta_n-\zeta_{n+1}+\zeta_3-\zeta_2)/(2-\zeta_2)$, with $\zeta_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of scaling relations including the ``bridge relation" for the scaling exponent of dissipation fluctuations $\mu=2-\zeta_6$.Comment: PRL, Submitted. REVTeX, 4 pages, I fig. (not included) PS Source of the paper with figure avalable at http://lvov.weizmann.ac.il/onlinelist.htm

Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence

The Lagrangian and Eulerian transversal velocity structure functions of fully developed fluid turbulence are found basing on the Navier-Stokes equation. The structure functions are shown to obey the scaling relations inside the inertial range. The scaling exponents are calculated analytically without using dimensional considerations. The obtained values are in a very good agreement with recent numerical and experimental data.Comment: 4 pages, 1 figur

Locality and stability of the cascades of two-dimensional turbulence

We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. A careful discussion of the subtle notion of locality is given at the end of the paper.Comment: v2: 23 pages; 4 figures; minor revisions; resubmitted to Phys. Rev.

The supercluster--void network III. The correlation function as a geometrical statistic

We investigate properties of the correlation function of clusters of galaxies using geometrical models. On small scales the correlation function depends on the shape and the size of superclusters. On large scales it describes the geometry of the distribution of superclusters. If superclusters are distributed randomly then the correlation function on large scales is featureless. If superclusters and voids have a tendency to form a regular lattice then the correlation function on large scales has quasi-regularly spaced maxima and minima of decaying amplitude; i.e., it is oscillating. The period of oscillations is equal to the step size of the grid of the lattice. We calculate the power spectrum for our models and compare the geometrical information of the correlation function with other statistics. We find that geometric properties (the regularity of the distribution of clusters on large scales) are better quantified by the correlation function. We also analyse errors in the correlation function and the power spectrum by generating random realizations of models and finding the scatter of these realizations.Comment: MNRAS LaTex style, 12 pages, 7 PostScript figures embedded, accepted by MNRA

Anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow

A model of the passive vector field advected by the uncorrelated in time Gaussian velocity with power-like covariance is studied by means of the renormalization group and the operator product expansion. The structure functions of the admixture demonstrate essential power-like dependence on the external scale in the inertial range (the case of an anomalous scaling). The method of finding of independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions. The constructed operator bases, which include the powers of the dissipation operator and the enstrophy operator, provide the possibility to calculate the exponents of the anomalous scaling.Comment: 9 pages, LaTeX2e(iopart.sty), submitted to J. Phys. A: Math. Ge

Directed Random Walk on the Lattices of Genus Two

The object of the present investigation is an ensemble of self-avoiding and directed graphs belonging to eight-branching Cayley tree (Bethe lattice) generated by the Fucsian group of a Riemann surface of genus two and embedded in the Pincar\'e unit disk. We consider two-parametric lattices and calculate the multifractal scaling exponents for the moments of the graph lengths distribution as functions of these parameters. We show the results of numerical and statistical computations, where the latter are based on a random walk model.Comment: 17 pages, 8 figure

Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions

In this paper we derive here, on the basis of the NS eqs. a set of fusion rules for correlations of velocity differences when all the separation are in the inertial interval. Using this we consider the standard hierarchy of equations relating the $n$-th order correlations (originating from the viscous term in the NS eq.) to $n+1$'th order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain is decoupled in the sense that the correlations of $n+1$'th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of eqs. when some separations go to zero we derive a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every n'th order correlation function of velocity differences {\cal F}_n(\B.R_1,\B.R_2,\dots) exhibits its own cross-over length $\eta_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. When all these separations are of the same order $R$ this length scales like $\eta_n(R)\sim \eta (R/L)^{x_n}$ with $x_n=(\zeta_n-\zeta_{n+1}+\zeta_3-\zeta_2)/(2-\zeta_2)$, with $\zeta_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents $\zeta_n$ of the $n$'th order structure functions. One of these relations is the well known ``bridge relation" for the scaling exponent of dissipation fluctuations $\mu=2-\zeta_6$.Comment: PRE, Submitted. REVTeX, 18 pages, 7 figures (not included) PS Source of the paper with figures avalable at http://lvov.weizmann.ac.il/onlinelist.htm

Dynamics of Passive-Scalar Turbulence

We present the first study of the dynamic scaling or multiscaling of passive-scalar and passive-vector turbulence. For the Kraichnan version of passive-scalar and passive-vector turbulence we show analytically, in both Eulerian and quasi-Lagrangian frameworks, that simple dynamic scaling is obtained but with different dynamic exponents. By developing the multifractal model we show that dynamic multiscaling occurs in passive-scalar turbulence only if the advecting velocity field is itself multifractal. We substantiate our results by detailed numerical simulations in shell models of passive-scalar advection.Comment: published versio

Hybridization-related correction to the jellium model for fullerenes

We introduce a new type of correction for a more accurate description of fullerenes within the spherically symmetric jellium model. This correction represents a pseudopotential which originates from the comparison between an accurate ab initio calculation and the jellium model calculation. It is shown that such a correction to the jellium model allows one to account, at least partly, for the sp2-hybridization of carbon atomic orbitals. Therefore, it may be considered as a more physically meaningful correction as compared with a structureless square-well pseudopotential which has been widely used earlier.Comment: 16 pages, 10 figure

Mean- Field Approximation and a Small Parameter in Turbulence Theory

Numerical and physical experiments on two-dimensional (2d) turbulence show that the differences of transverse components of velocity field are well described by a gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume existence of critical space-dimensionality $d=d_{c}$, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At $d<d_{c}$ the flow can be described by the ``mean-field theory'', leading to the observed gaussian statistics and Kolmogorov scaling of transverse velocity differences. It is shown that in the vicinity of $d=d_{c}$ the ratio of the relaxation and translation characteristic times decreases to zero, thus giving rise to a small parameter of the theory. The expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of $d=d_{c}$. The resulting equation describes experimental data on two-dimensional turbulence and demonstrate onset of intermittency as $d-d_{c}>0$ and $r/L\to 0$ in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity difference in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.Comment: 25 pages, 1 figur