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

Abstract

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.

43 pages, revtex3

References in corpus (6)

• Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution
• Exact Resummations in the Theory of Hydrodynamic Turbulence: I The Ball of Locality and Normal Scaling
• Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling
• "Intermittency" in Hydrodynamic Turbulence as Intermediate Asymptotic to Kolmogorov41 Scaling
• Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency
• Anomalous Scaling and Fusion Rules in Hydrodynamic Turbulence

Cited by in corpus (34)

• ANOMALOUS SCALING OF THE PASSIVE SCALAR
• Anisotropy in Turbulent Flows and in Turbulent Transport
• Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar
• Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence
• Renormalization Group, Operator Product Expansion, and Anomalous Scaling in a Model of Advected Passive Scalar
• Anomalous Scaling in the N-Point Functions of Passive Scalar
• Intermittency in passive scalar advection
• Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field
• Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow
• Fully developed isotropic turbulence: nonperturbative renormalization group formalism and fixed point solution
• Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions
• Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ${\epsilon }^3$
• Intermittency and Anomalous Scaling of Passive Scalars in Any Space Dimension
• Passive Scalar: Scaling Exponents and Realizability
• Universality in Turbulence: an Exactly Soluble Model
• Anomalous scaling in two models of the passive scalar advection: Effects of anisotropy and compressibility
• Energy and Vorticity Spectra in Turbulent Superfluid ${}^4$He from $T=0$ to $T_{\lambda }$
• Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multi-scaling
• Analytic Calculation of the Anomalous Exponents in Turbulence: Using the Fusion Rules to Flush Out a Small Parameter
• Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency
• Analytic calculation of anomalous scaling in random shell models for a passive scalar
• The Universal Scaling Exponents of Anisotropy in Turbulence and their Measurement
• Anomalous scaling in random shell models for passive scalars
• The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions
• Direct Numerical Simulations of the Kraichnan Model: Scaling Exponents and Fusion Rules
• Statistical symmetry restoration in fully developed turbulence: Renormalization group analysis of two models
• Fusion Rules in Navier-Stokes Turbulence: First Experimental Tests
• The passive polymer problem
• On Conditional Statistics in Scalar Turbulence: Theory vs. Experiment
• Functional renormalization-group approach to decaying turbulence
• Scaling Behavior in Turbulence is Doubly Anomalous
• Turbulence under a magnifying glass
• Explorations into the inertial and integral scales of homogeneous axisymmetric turbulence
• Heat equilibrium distribution in a turbulent flow