Chaotic Cascades with Kolmogorov 1941 Scaling

TL;DR

This paper introduces a deterministic chaotic variant of turbulence cascade models that maintains Kolmogorov 1941 scaling by avoiding large deviations, supported by numerical evidence and theoretical analysis.

Contribution

It presents a new deterministic chaotic cascade model that preserves K41 scaling, contrasting with traditional random models affected by large deviations.

Findings

Deterministic models can be chaotic and still exhibit exact K41 scaling.

A mechanism involving neutrally unstable fixed points helps avoid large deviations.

Numerical studies support the theoretical predictions.

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 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.