The vanishing viscosity limit for a dyadic model

TL;DR

This paper investigates a dyadic shell model related to turbulence, demonstrating that as viscosity approaches zero, the system's behavior converges to that of the inviscid case, supporting the dissipation anomaly concept.

Contribution

It proves the convergence of the viscous system's attractor to the inviscid system's attractor as viscosity diminishes, linking to Kolmogorov's dissipation anomaly theory.

Findings

Unique fixed point is a global attractor.

Convergence of dissipation rates as viscosity tends to zero.

Supports Kolmogorov's dissipation anomaly in a shell model.

Abstract

A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov's theory for the actual Navier-Stokes equations.