Intermittency and lower dimensional dissipation in incompressible fluids

TL;DR

This paper demonstrates that in inviscid incompressible fluids, lower-dimensional energy dissipation leads to deviations from classical turbulence predictions, providing a rigorous foundation for intermittency and the fractal nature of turbulent dissipation structures.

Contribution

It establishes a rigorous link between dissipation dimensionality and structure function deviations, connecting geometrical dissipation assumptions with turbulence intermittency.

Findings

Lower-dimensional dissipation implies deviations from K41 predictions for p>3.

The upper bound on structure function exponents matches the β-model from 1970s.

A new local variant of Constantin-E-Titi argument is developed for the proof.

Abstract

In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as ``intermittency'' and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents $\zeta_p=\frac{p}{3}$ might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every $p-$th order structure function for $p>3$. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor's frozen turbulence hypothesis, our strongest upper bound on $\zeta_p$ coincides with the $\beta-$model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the $p-$th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.