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Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect

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Abstract

This paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors. Three important topics are analyzed with complete mathematical rigor for this model: (1) Renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces, (2) the effect of non-Gaussian turbulent velocity statistics on renormalization theory, (3) the “sweeping” effect of additional large scale mean velocities. A special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the Kolmogorov-spectrum in the model. In the authors' earlier paper, it was established that the Kolmogorov value is at a phase transition boundary in the exact renormalization theory. It is found here that the qualitative model, despite its simplicity contains, in the vicinity of the Kolmogorov value, a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories. In particular, the Richardson 4/3-law for pair dispersion and interfaces with fractal dimension defect of 2/3 occur in the model rigorously as limits when the Kolmogorov spectrum is approached as a limit from one side of the phase transition boundary; alternative corrections to the Richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1/3, occur in the model when the Kolmogorov spectrum is approached from the other side of the phase transition. It is very interesting that fractal dimension defects of roughly the value either 1/3 or 2/3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments. As regards non-Gaussian the asymptotic normality of normalized integrals (B.56) corresponding to compactly supported blobs with mean zero. The proof of this latter fact is done in the same way as Step 2, Proposition B.3, using the fact that the corresponding random processes\(\tilde V_\delta (s)\) have finite domain of dependence. This concludes the proof of Proposition B.4.

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Authors and Affiliations

  1. Courant Institute, New York University, 10012, New York, NY, USA

    Marco Avellaneda

  2. Department of Mathematics and Program in Appl. & Computa. Math., Princeton University, 08544, Princeton, NJ, USA

    Andrew Majda

Authors
  1. Marco Avellaneda
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  2. Andrew Majda
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Additional information

Communicated by S.-T. Yau

Research partially supported by NSF-DMS-9005799, ARO-DAAL03-89-K-0039 and AFOSR 90-0090

Research partially supported by grants NSF-DMS-90-01805, ARO-DAAL03-89-K-0013, and ONR-N00014-89-J-1014

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Avellaneda, M., Majda, A. Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect. Commun.Math. Phys. 146, 139–204 (1992). https://doi.org/10.1007/BF02099212

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