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

Foias, Holm & Titi \cite{FHT2} have settled the problem of existence and uniqueness for the 3D \lans equations on periodic box $[0,L{]}^3$. There still remains the problem, first introduced by Doering and Foias \cite{DF} for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number $\text{Rey}$, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. $\text{Rey}$ is defined as $\text{Rey}=U\mathrm{\ell }\mathrm{/}\nu$ where $U$ is a bounded spatio-temporally averaged Navier-Stokes velocity field and $\mathrm{\ell }$ the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by $\mathrm{\ell }{\lambda }_k^{-1}\le c(\mathrm{\ell }\mathrm{/}\alpha {)}^{1\mathrm{/}4}{\text{Rey}}^{5\mathrm{/}8}$. Moreover, the estimate of Foias, Holm & Titi for the fractal dimension of the global attractor, in terms of $\text{Rey}$, comes out to be $$d_F(\mathcal{A})\le c\frac{V_{\alpha }{V}_{\mathrm{\ell }}^{1\mathrm{/}2}}{(L^2{\lambda }_1{)}^{9\mathrm{/}8}}{\text{Rey}}^{9\mathrm{/}4}$$ where $V_{\alpha }=(L\mathrm{/}(\mathrm{\ell }\alpha {)}^{1\mathrm{/}2}{)}^3$ and $V_{\mathrm{\ell }}=(L\mathrm{/}\mathrm{\ell }{)}^3$. It is also shown that there exists a series of time-averaged inverse squared length scales whose members, $<{\kappa }_{n,0}^2>$, %, are related to the $2n$th-moments of the energy spectrum when $\alpha \to 0$. are estimated as $(n\ge 1)$ $${\mathrm{\ell }}^2<{\kappa }_{n,0}^2>\le c_{n,\alpha }{V}_{\alpha }^{\frac{n-1}{n}}{\text{Rey}}^{11\mathrm{/}4-\frac{7}{4n}}(\ln \text{Rey}{)}^{\frac{1}{n}}+c_1\text{Rey}(\ln \text{Rey})\mathrm{.}$$ The upper bound on the first member of the hierarchy $<{\kappa }_{1,0}^2>$ coincides with the inverse squared Taylor micro-scale to within log-corrections.