Kolmogorov's dissipation number and determining wavenumber for dyadic models

TL;DR

This paper investigates dyadic models for fluid dynamics, establishing fixed points, stability, and a relationship between a time-dependent determining wavenumber and Kolmogorov's dissipation wavenumber, supported by numerical simulations.

Contribution

It introduces a new analysis of the fixed point stability and the relationship between the determining wavenumber and Kolmogorov's dissipation wavenumber in dyadic models.

Findings

Existence and stability of fixed points in dyadic models.

The time-averaged determining wavenumber is bounded by Kolmogorov's dissipation wavenumber.

Numerical simulations confirm the energy spectrum behavior below the dissipation wavenumber.

Abstract

We study some dyadic models for incompressible magnetohydrodynamics and Navier-Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov's dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov's dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov's dissipation wavenumber.