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

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

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.

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

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

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

Statistical model for intermittent plasma edge turbulence

The Probability Distribution Function of plasma density fluctuations at the edge of fusion devices is known to be skewed and strongly non-Gaussian. The causes of this peculiar behaviour are, up to now, largely unexplored. On the other hand, understanding the origin and the properties of edge turbulence is a key issue in magnetic fusion research. In this work we show that a stochastic fragmentation model, already successfully applied to fluid turbulence, is able to predict an asymmetric distribution that closely matches experimental data. The asymmetry is found to be a direct consequence of intermittency. A discussion of our results in terms of recently suggested BHP universal curve [S.T. Bramwell, P.C.W. Holdsworth, J.-F. Pinton, Nature (London) 396, 552 (1998)], that should hold for strongly correlated and critical systems, is also proposedComment: 13 pages. Physica Review E, accepte

On the intermittent energy transfer at viscous scales in turbulent flows

In this letter we present numerical and experimental results on the scaling properties of velocity turbulent fields in the range of scales where viscous effects are acting. A generalized version of Extended Self Similarity capable of describing scaling laws of the velocity structure functions down to the smallest resolvable scales is introduced. Our findings suggest the absence of any sharp viscous cutoff in the intermittent transfer of energy.Comment: 10 pages, plain Latex, 6 figures available upon request to [email protected]