Analytic Calculation of the Anomalous Exponents in Turbulence: Using the Fusion Rules to Flush Out a Small Parameter

Abstract

The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorov's statistics of 1941 acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents ${\zeta }_n$ of the $n$th order structure functions can be computed analytically. The crux of the method consists of renormalizing a 4-point interaction amplitude on the basis of the fusion rules. The novelty is that this amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of ${\zeta }_2$, ${\delta }_2={\zeta }_2-2/3\approx 0.03$. Higher order interaction amplitudes are shown to be even smaller. The corrections to K41 to $O({\delta }_2)$ result from standard logarithmically divergent ladder-diagrams in which the 4-point interaction acts as a "rung". The theory allows a calculation of the anomalous exponents ${\zeta }_n$ in powers of the small parameter ${\delta }_2$. The $n$-dependence of the scaling exponents ${\zeta }_n$ stems from pure combinatorics of the ladder diagrams. We calculate the exponents ${\zeta }_n$ up to $O({\delta }_2^3)$. Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter ${\delta }_2$. The actual evaluation of the small parameter ${\delta }_2$ from first principles requires additional developments that are outside the scope of this paper.

PRE submitted, 23 pages, 11 figures included

References in corpus (3)

• 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
• Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multi-scaling

Cited by in corpus (21)

• Fully developed isotropic turbulence: nonperturbative renormalization group formalism and fixed point solution
• Discontinuous transition from direct to inverse cascade in three-dimensional turbulence
• Probing small-scale intermittency with a fluctuation theorem
• Testing for Markovian Character and Modeling of Intermittency in Solar Wind Turbulence
• Lagrangian structure functions in fully-developed hydrodynamical turbulence
• Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum
• The State of the Art in Hydrodynamic Turbulence: Past Successes and Future Challenges
• On the elimination of the sweeping interactions from theories of hydrodynamic turbulence
• On the Anomalous Scaling Exponents in Nonlinear Models of Turbulence
• Fluctuation-response relation in turbulent systems
• Generation of small-scale structures in the developed turbulence
• Shell model intermittency is the hidden self-similarity
• Hidden scale invariance of turbulence in a shell model: from forcing to dissipation scales
• Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics
• Locality and stability of the cascades of two-dimensional turbulence
• Odd viscosity suppresses intermittency in direct turbulent cascades
• Dissipation scales and anomalous sinks in steady two-dimensional turbulence
• Yakhot's model of strong turbulence: A generalization of scaling models of turbulence
• Winterberg's conjectured breaking of the superluminal quantum correlations over large distances
• Particle transport in a random velocity field with Lagrangian statistics
• Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence