  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.

Fluid Dynamics;Analysis of PDEs