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

Foias, Holm & Titi [15] have settled the problem of existence and uniqueness for the 3D LANS-α equations on periodic box [0, L]. There still remains the problem, first introduced by Doering and Foias [17] for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number Re, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. Re is defined as Re = 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 lλ k ≤ c (l/α) Re. Moreover, the estimate of Foias, Holm & Titi for the fractal dimension of the global attractor, in terms of Re, comes out to be