L. Biferale, M. Blank and U. Frisch
Chaotic cascades with Kolmogorov 1941 scaling
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ABSTRACT. We define a (chaotic) deterministic variant of random
multiplicative cascade models of turbulence. It preserves the
hierarchical tree structure, thanks to the addition of
infinitesimal noise. The zero-noise limit can be handled by
Perron-Frobenius theory, just as the zero-diffusivity limit for the
fast dynamo problem. Random multiplicative models do not possess
Kolmogorov 1941 (K41) scaling because of a large-deviations effect.
Our numerical studies indicate that {\it deterministic}
multiplicative models can be chaotic and still have exact K41
scaling. A mechanism is suggested for avoiding large deviations,
which is present in maps with a neutrally unstable fixed point.