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Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics

I. Arad·I. Procaccia
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Abstract

The problem of anomalous scaling in passive scalar advection, especially with $δ$-correlated velocity field (the Kraichnan model) has attracted a lot of interest since the exponents can be computed analytically in certain limiting cases. In this paper we focus, rather than on the evaluation of the exponents, on elucidating the {\em physical mechanism} responsible for the anomaly. We show that the anomalous exponents $ζ_n$ stem from the Lagrangian dynamics of shapes which characterize configurations of n points in space. Using the shape-to-shape transition probability, we define an operator whose eigenvalues determine the anomalous exponents for all n, in all the sectors of the SO(3) symmetry group.

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I. Arad, I. Procaccia. 2000-01-30. Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics. https://arxiv.org/abs/nlin/0001067

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