Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2001-06-15
Phys. Rev. E64: 056306 (2001)
Nonlinear Sciences
Chaotic Dynamics
34 pages, 5 figures, REVTeX
Scientific paper
10.1103/PhysRevE.64.056306
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 $\propto\delta(t-t')/k^{d+\eps}$. 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.
Adzhemyan Loran Ts.
Antonov Nikolaj V.
Barinov V. A.
Kabrits Yu. S.
Vasil'ev Alexander N.
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