607 research outputs found

### Multiscaling in passive scalar advection as stochastic shape dynamics

The Kraichnan rapid advection model is recast as the stochastic dynamics of
tracer trajectories. This framework replaces the random fields with a small set
of stochastic ordinary differential equations. Multiscaling of correlation
functions arises naturally as a consequence of the geometry described by the
evolution of N trajectories. Scaling exponents and scaling structures are
interpreted as excited states of the evolution operator. The trajectories
become nearly deterministic in high dimensions allowing for perturbation theory
in this limit. We calculate perturbatively the anomalous exponent of the third
and fourth order correlation functions. The fourth order result agrees with
previous calculations.Comment: 14 pages, LaTe

### Non-universality of the scaling exponents of a passive scalar convected by a random flow

We consider passive scalar convected by multi-scale random velocity field
with short yet finite temporal correlations. Taking Kraichnan's limit of a
white Gaussian velocity as a zero approximation we develop perturbation theory
with respect to a small correlation time and small non-Gaussianity of the
velocity. We derive the renormalization (due to temporal correlations and
non-Gaussianity) of the operator of turbulent diffusion. That allows us to
calculate the respective corrections to the anomalous scaling exponents of the
scalar field and show that they continuously depend on velocity correlation
time and the degree of non-Gaussianity. The scalar exponents are thus non
universal as was predicted by Shraiman and Siggia on a phenomenological ground
(CRAS {\bf 321}, 279, 1995).Comment: 4 pages, RevTex 3.0, Submitted to Phys.Rev.Let

### Anomalous Scaling in a Model of Passive Scalar Advection: Exact Results

Kraichnan's model of passive scalar advection in which the driving velocity
field has fast temporal decorrelation is studied as a case model for
understanding the appearance of anomalous scaling in turbulent systems. We
demonstrate how the techniques of renormalized perturbation theory lead (after
exact resummations) to equations for the statistical quantities that reveal
also non perturbative effects. It is shown that ultraviolet divergences in the
diagrammatic expansion translate into anomalous scaling with the inner length
acting as the renormalization scale. In this paper we compute analytically the
infinite set of anomalous exponents that stem from the ultraviolet divergences.
Notwithstanding, non-perturbative effects furnish a possibility of anomalous
scaling based on the outer renormalization scale. The mechanism for this
intricate behavior is examined and explained in detail. We show that in the
language of L'vov, Procaccia and Fairhall [Phys. Rev. E {\bf 50}, 4684 (1994)]
the problem is ``critical" i.e. the anomalous exponent of the scalar primary
field $\Delta=\Delta_c$. This is precisely the condition that allows for
anomalous scaling in the structure functions as well, and we prove that this
anomaly must be based on the outer renormalization scale. Finally, we derive
the scaling laws that were proposed by Kraichnan for this problem, and show
that his scaling exponents are consistent with our theory.Comment: 43 pages, revtex

### Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar

For a delta-correlated velocity field, simultaneous correlation functions of
a passive scalar satisfy closed equations. We analyze the equation for the
four-point function. To describe a solution completely, one has to solve the
matching problems at the scale of the source and at the diffusion scale. We
solve both the matching problems and thus find the dependence of the four-point
correlation function on the diffusion and pumping scale for large space
dimensionality $d$. It is shown that anomalous scaling appears in the first
order of $1/d$ perturbation theory. Anomalous dimensions are found analytically
both for the scalar field and for it's derivatives, in particular, for the
dissipation field.Comment: 19 pages, RevTex 3.0, Submitted to Phys.Rev. E, revised versio

### Rain, power laws, and advection

Localized rain events have been found to follow power-law size and duration
distributions over several decades, suggesting parallels between precipitation
and seismic activity [O. Peters et al., PRL 88, 018701 (2002)]. Similar power
laws are generated by treating rain as a passive tracer undergoing advection in
a velocity field generated by a two-dimensional system of point vortices.Comment: 7 pages, 4 figure

### Fusion Rules in Turbulent Systems with Flux Equilibrium

Fusion rules in turbulence specify the analytic structure of many-point
correlation functions of the turbulent field when a group of coordinates
coalesce. We show that the existence of flux equilibrium in fully developed
turbulent systems combined with a direct cascade induces universal fusion
rules. In certain examples these fusion rules suffice to compute the
multiscaling exponents exactly, and in other examples they give rise to an
infinite number of scaling relations that constrain enormously the structure of
the allowed theory.Comment: Submitted to PRL on July 95, 4 pages, REVTe

### On Conditional Statistics in Scalar Turbulence: Theory vs. Experiment

We consider turbulent advection of a scalar field T(\B.r), passive or
active, and focus on the statistics of gradient fields conditioned on scalar
differences $\Delta T(R)$ across a scale $R$. In particular we focus on two
conditional averages $\langle\nabla^2 T\big|\Delta T(R)\rangle$ and
$\langle|\nabla T|^2\big|\Delta T(R) \rangle$. We find exact relations between
these averages, and with the help of the fusion rules we propose a general
representation for these objects in terms of the probability density function
$P(\Delta T,R)$ of $\Delta T(R)$. These results offer a new way to analyze
experimental data that is presented in this paper. The main question that we
ask is whether the conditional average $\langle\nabla^2 T\big| \Delta
T(R)\rangle$ is linear in $\Delta T$. We show that there exists a dimensionless
parameter which governs the deviation from linearity. The data analysis
indicates that this parameter is very small for passive scalar advection, and
is generally a decreasing function of the Rayleigh number for the convection
data.Comment: Phys. Rev. E, Submitted. REVTeX, 10 pages, 5 figs. (not included) PS
Source of the paper with figure available at
http://lvov.weizmann.ac.il/onlinelist.html#unpub

### Quasi-Gaussian Statistics of Hydrodynamic Turbulence in 3/4+\epsilon dimensions

The statistics of 2-dimensional turbulence exhibit a riddle: the scaling
exponents in the regime of inverse energy cascade agree with the K41 theory of
turbulence far from equilibrium, but the probability distribution functions are
close to Gaussian like in equilibrium. The skewness \C S \equiv
S_3(R)/S^{3/2}_2(R) was measured as \C S_{\text{exp}}\approx 0.03. This
contradiction is lifted by understanding that 2-dimensional turbulence is not
far from a situation with equi-partition of enstrophy, which exist as true
thermodynamic equilibrium with K41 exponents in space dimension of $d=4/3$. We
evaluate theoretically the skewness \C S(d) in dimensions ${4/3}\le d\le 2$,
show that \C S(d)=0 at $d=4/3$, and that it remains as small as \C
S_{\text{exp}} in 2-dimensions.Comment: PRL, submitted, REVTeX 4, 4 page

### Turbulence Fluctuations and New Universal Realizability Conditions in Modelling

General turbulent mean statistics are shown to be characterized by a
variational principle. The variational functionals, or ``effective actions'',
have experimental consequences for turbulence fluctuations and are subject to
realizability conditions of positivity and convexity. An efficient
Rayleigh-Ritz algorithm is available to calculate approximate effective actions
within PDF closures. Examples are given for Navier-Stokes and for a 3-mode
system of Lorenz. The new realizability conditions succeed at detecting {\em a
priori} the poor predictions of PDF closures even when the classical 2nd-order
moment realizability conditions are satisfied.Comment: 4 pages, LaTeX (Version 2.09), 3 figures, Postscript, Submitted to
Phys. Rev. Let

### Dynamical scaling and intermittency in shell models of turbulence

We introduce a model for the turbulent energy cascade aimed at studying the
effect of dynamical scaling on intermittency. In particular, we show that by
slowing down the energy transfer mechanism for fixed energy flux, intermittency
decreases and eventually disappears. This result supports the conjecture that
intermittency can be observed only if energy is flowing towards faster and
faster scales of motion.Comment: 4 pages, 3 figure

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