Length-scale estimates for the LANS-αα 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-αα equations on periodic box [0,L]3[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 ReRe, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. ReRe is defined as Re=Uℓ/νRe=Uℓ/ν where UU is a bounded spatio-temporally averaged Navier–Stokes velocity field and ℓℓ the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by ℓλk−1≤c(ℓ/α)1/4Re5/8. Moreover, the estimate of Foias, Holm and Titi for the fractal dimension of the global attractor, in terms of ReRe, comes out to be dF(A)≤cVαVℓ1/2(L2λ1)9/8Re9/4 where Vα=(L/(ℓα)1/2)3Vα=(L/(ℓα)1/2)3 and Vℓ=(L/ℓ)3Vℓ=(L/ℓ)3. It is also shown that there exists a series of time-averaged inverse squared length scales whose members, 〈κn,02〉, are estimated as (n≥1)(n≥1)ℓ2〈κn,02〉≤cn,αVαn−1nRe114−74n(lnRe)1n+c1Re(lnRe). The upper bound on the first member of the hierarchy 〈κ1,02〉 coincides with the inverse squared Taylor micro-scale to within log-corrections.