Kolmogorov's dissipation number and determining wavenumber for dyadic models

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.

References in corpus (8)

• An optimal shell model of turbulence
• Shell Models of Magnetohydrodynamic Turbulence
• Analytic Study of Shell Models of Turbulence
• The vanishing viscosity limit for a dyadic model
• Smooth solutions for the dyadic model
• Determining modes for the 3D Navier-Stokes equations
• Uniqueness and non-uniqueness results for dyadic MHD models
• Blow-up of dyadic MHD models with forward energy cascade