Blowup as a driving mechanism of turbulence in shell models

TL;DR

This paper demonstrates that finite-time blowup events drive turbulence in shell models, with instantons forming coherent structures that influence energy cascades and anomalous scaling, supported by analytical and numerical evidence.

Contribution

It establishes a direct link between blowup phenomena and turbulence mechanisms in shell models, revealing universal self-similar instanton structures.

Findings

Blowups generate instantons that travel through inertial range.

Instantons are described by universal self-similar statistics.

Anomalous scaling is analytically connected to instanton creation.

Abstract

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949 Onsager noticed that the scaling properties in inertial range imply non-differentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.