On the asymptotic behavior of average energy and enstrophy in 3D turbulent flows

Rigorous upper and lower bounds are proved for the Taylor and the Kolmogorov wavenumbers for the three-dimensional space periodic Navier–Stokes equations. Under the assumption that Kolmogorov’s two-thirds power law holds, the bounds sharpen to κ T ∼ Gr 1 / 4 and κ ϵ ∼ Gr 3 / 8 respectively, where Gr is the Grashof number. This provides a rigorous proof that the power law implies (1) the energy cascade, (2) Kolmogorov dissipation law, and (3) a connection between κ T and κ ϵ . The portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite-time average.