Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ${\epsilon }^3$

Abstract

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function $\proptoδ(t-t')/k^{d+\eps}$. Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the $\epsilon$ expansion to order ${\epsilon }^3$ (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the $\epsilon$ expansion are discussed; the improved ``inverse'' $\epsilon$ expansion is proposed and the comparison with the existing nonperturbative results is given.

34 pages, 5 figures, REVTeX

References in corpus (2)

• Anomalous scaling, nonlocality and anisotropy in a model of the passively advected vector field
• Pressure and intermittency in passive vector turbulence

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