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

L. Ts. Adzhemyan·N. V. Antonov·V. A. Barinov·Yu. S. Kabrits·A. N. Vasil'ev
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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 $\varepsilon$ expansion to order $\varepsilon^{3}$ (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 $\varepsilon$ expansion are discussed; the improved ``inverse'' $\varepsilon$ expansion is proposed and the comparison with the existing nonperturbative results is given.

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L. Ts. Adzhemyan, N. V. Antonov, V. A. Barinov, Yu. S. Kabrits, A. N. Vasil'ev. 2001-06-15. Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order $\varepsilon^{3}$. https://doi.org/10.1103/physreve.64.056306

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