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Finite size Lyapunov exponent for some simple models of turbulence

Leonid Piterbarg
University of Southern California, Department of Mathematics
(Abstract received 03/14/2012 for session D)
ABSTRACT

In recent years FSLE has been successfully applied to variety of problems in physical oceanography and physics of atmosphere. In particular, the approach was used for detecting barriers to transport, identifying hyperbolic manifolds, measuring local stirring, identifying Lagrangian coherent structures, and studying biological activity in upwelling systems. To better interpret FSLE and properly calibrate the corresponding computational algorithms exactly solvable models would be of great importance.

We present and analyze explicit FSLE expressions for several idealized models of turbulence in 1D and 2D. Among them are a random walk with discrete time and continuously distributed jumps and an isotropic Brownian flow in 2D also known as the Kraichnan flow. For the former a surprising fact is a '-1' scaling for small separations in contrast to '-2' well known for a random walk in continuous time (Brownian flow) and for a simple random walk in discrete time. For the Kraichnan flow an exact relation is established between the scaling of FSLE and the scaling of relative dispersion in time.