Anomalous scaling in a model of passive scalar advection: Exact results

Adrienne L. Fairhall, Omri Gat, Victor L’vov, Itamar Procaccia

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Kraichnan’s model of passive scalar advection in which the driving velocity field has fast temporal decorrelation is studied as a case model for understanding the appearance of anomalous scaling in turbulent systems. We demonstrate how the techniques of renormalized perturbation theory lead (after exact resummations) to equations for the statistical quantities that also reveal nonperturbative effects. It is shown that ultraviolet divergences in the diagrammatic expansion translate into anomalous scaling with the inner length acting as the renormalization scale. In this paper, we compute analytically the infinite set of anomalous exponents that stem from the ultraviolet divergences. Notwithstanding these computations, nonperturbative effects furnish a possibility of anomalous scaling based on the outer renormalization scale. The mechanism for this intricate behavior is examined and explained in detail. We show that in the language of L’vov, Procaccia, and Fairhall [Phys. Rev. E 50, 4684, (4684)], the problem is "critical," i.e., the anomalous exponent of the scalar primary field [Formula Presented]. This is precisely the condition that allows for anomalous scaling in the structure functions as well, and we prove that this anomaly must be based on the outer renormalization scale. Finally, we derive the scaling laws that were proposed by Kraichnan for this problem and show that his scaling exponents are consistent with our theory.

Original languageAmerican English
Pages (from-to)3518-3535
Number of pages18
JournalPhysical Review E
Issue number4
StatePublished - 1996
Externally publishedYes


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