Length-scale estimates for the LANS- α  equations in terms of the Reynolds number

Foias, Holm and Titi [C. Foias, D.D. Holm, E.S. Titi, The three dimensional viscous Camassa–Holm equations and their relation to the Navier–Stokes equations and turbulence theory, J. Dynam. Differential Equations 14 (2002) 1–35] have settled the problem of existence and uniqueness for the 3D LANS- α equations on periodic box [ 0 , L ] 3 . There still remains the problem, first introduced by Doering and Foias [C.R. Doering, C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech. 467 (2002) 289–306] for the Navier–Stokes equations, of obtaining estimates in terms of the Reynolds number R e , whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. R e is defined as R e = U ℓ / ν where U is a bounded spatio-temporally averaged Navier–Stokes velocity field and ℓ the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by ℓ λ k − 1 ≤ c ( ℓ / α ) 1 / 4 R e 5 / 8 . Moreover, the estimate of Foias, Holm and Titi for the fractal dimension of the global attractor, in terms of R e , comes out to be d F ( A ) ≤ c V α V ℓ 1 / 2 ( L 2 λ 1 ) 9 / 8 R e 9 / 4 where V α = ( L / ( ℓ α ) 1 / 2 ) 3 and V ℓ = ( L / ℓ ) 3 . It is also shown that there exists a series of time-averaged inverse squared length scales whose members, 〈 κ n , 0 2 〉 , are estimated as ( n ≥ 1 ) ℓ 2 〈 κ n , 0 2 〉 ≤ c n , α V α n − 1 n R e 11 4 − 7 4 n ( ln R e ) 1 n + c 1 R e ( ln R e ) . The upper bound on the first member of the hierarchy 〈 κ 1 , 0 2 〉 coincides with the inverse squared Taylor micro-scale to within log-corrections.