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

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Foias, Holm & Titi FHT2 have settled the problem of existence and uniqueness for the 3D equations on periodic box [0,L]³. There still remains the problem, first introduced by Doering and Foias DF for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. is defined as = Uℓ/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 ℓλₖ⁻¹ ≤ c (ℓ/α)¹/⁴⁵/⁸. Moreover, the estimate of Foias, Holm & Titi for the fractal dimension of the global attractor, in terms of, comes out to be dF(A) ≤ c VαVℓ¹/²(L²λ₁)⁹/⁸ ⁹/⁴ where Vα = (L/(ℓα)¹/²)³ and Vℓ = (L/ℓ)³. It is also shown that there exists a series of time-averaged inverse squared length scales whose members, <κₙ,₀²>, %, are related to the 2nth-moments of the energy spectrum when α→ 0. are estimated as (n≥ 1) ℓ²<κₙ,₀²> ≤ cₙ,αVαⁿ⁻¹/ⁿ 11/4 - 7/4n()¹/ⁿ + c₁(). The upper bound on the first member of the hierarchy <κ₁,₀²> coincides with the inverse squared Taylor micro-scale to within log-corrections.