Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics

Details Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics

2000-01-30

arxiv.org/abs/nlin/0001067v1

Nonlinear Sciences

Chaotic Dynamics

6 pages, IUTAM simposium, T.Kambe, ed. in press

Scientific paper

The problem of anomalous scaling in passive scalar advection, especially with $\delta$-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 $\zeta_n$ 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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