On the Field Theoretical Approach to the Anomalous Scaling in Turbulence

TL;DR

This paper applies a field theoretical approach to analyze anomalous scaling in turbulence, focusing on specific dimensions and identifying key operators that explain the scaling behavior.

Contribution

It extends the field theoretical framework to the Navier-Stokes equation, demonstrating the role of composite operators with negative dimensions in anomalous scaling.

Findings

Infinite set of Galilean invariant operators with negative dimensions confirmed.

Explicit expression for the junior operator related to energy dissipation derived.

Critical dimension of the junior operator varies with space dimension, being strongly negative in 2D.

Abstract

Anomalous scaling problem in the stochastic Navier-Stokes equation is treated in the framework of the field theoretical approach, successfully applied earlier to the Kraichnan rapid advection model. Two cases of the space dimensions d=2 and d->infinity, which allow essential simplification of the calculations, are analysed. The presence of infinite set of the Galilean invariant composite operators with negative critical dimensions in the model discussed has been proved. It allows, as well as for the Kraichnan model, to justify the anomalous scaling of the structure functions. The explicit expression for the junior operator of this set, related to the square of energy dissipation operator, has been found in the first order of the epsilon-expansion. Its critical dimension is strongly negative in the two dimensional case and vanishes while d->infinity.