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

Passive scalar convection in 2D long-range delta-correlated velocity field: Exact results

The letter presents new field-theoretical approach to 2D passive scalar problem. The Gaussian form of the distribution for the Lyapunov exponent is derived and its parameters are found explicitly.Comment: 11 pages, RevTex 3.0, IFUM-94/455/January-F

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

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

A Simple Passive Scalar Advection-Diffusion Model

This paper presents a simple, one-dimensional model of a randomly advected passive scalar. The model exhibits anomalous inertial range scaling for the structure functions constructed from scalar differences. The model provides a simple computational test for recent ideas regarding closure and scaling for randomly advected passive scalars. Results suggest that high order structure function scaling depends on the largest velocity eddy size, and hence scaling exponents may be geometry-dependent and non-universal.Comment: 30 pages, 11 figure

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

Excitation of stellar p-modes by turbulent convection: 1. Theoretical formulation

Stochatic excitation of stellar oscillations by turbulent convection is investigated and an expression for the power injected into the oscillations by the turbulent convection of the outer layers is derived which takes into account excitation through turbulent Reynolds stresses and turbulent entropy fluctuations. This formulation generalizes results from previous works and is built so as to enable investigations of various possible spatial and temporal spectra of stellar turbulent convection. For the Reynolds stress contribution and assuming the Kolmogorov spectrum we obtain a similar formulation than those derived by previous authors. The entropy contribution to excitation is found to originate from the advection of the Eulerian entropy fluctuations by the turbulent velocity field. Numerical computations in the solar case in a companion paper indicate that the entropy source term is dominant over Reynold stress contribution to mode excitation, except at high frequencies.Comment: 14 pages, accepted for publication in A&

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