Structure function and fractal dissipation for an intermittent inviscid dyadic model

TL;DR

This paper explores a generalized dyadic model for turbulence, revealing multifractal structures and anomalous energy dissipation on fractal sets, aligning with experimental observations.

Contribution

It introduces a variable coefficient dyadic model with intermittency, providing explicit analysis of its multifractal structure and energy dissipation properties.

Findings

The structure function exponent is concave, matching theoretical and experimental models.

Anomalous energy dissipation occurs on a fractal set with dimension less than 3.

The model exhibits rich multifractal behavior with a well-defined fixed point.

Abstract

We study a generalization of the original tree-indexed dyadic model by Katz and Pavlovi\'c for the turbulent energy cascade of three-dimensional Euler equation. We allow the coefficients to vary with some restrictions, thus giving the model a realistic spatial intermittency. By introducing a forcing term on the first component, the fixed point of the dynamics is well defined and some explicit computations allow to prove the rich multifractal structure of the solution. In particular the exponent of the structure function is concave in accordance with other theoretical and experimental models. Moreover anomalous energy dissipation happens in a fractal set of dimension strictly less than 3.