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The tradition in Navier-Stokes analysis of finding estimates in terms of theGrashof number $\bG$, whose character depends on the ratio of the forcing tothe viscosity $\nu$, means that it is difficult to make comparisons with otherresults expressed in terms of Reynolds number $\Rey$, whose character dependson the fluid response to the forcing. The first task of this paper is to applythe approach of Doering and Foias \cite{DF} to the two-dimensionalNavier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimatingquantities of physical relevance, particularly long-time averages$\left<\cdot\right>$, in terms of the Reynolds number $\Rey = U\ell/\nu$, where$U^{2}= L^{-2}\left<\|\bu\|_{2}^{2}\right>$ and $\ell$ is the forcing scale. Inparticular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractordimension converts to $a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}$, while the estimatefor the inverse Kraichnan length is $(a_{\ell}^{2}\Rey)^{1/2}$, where$a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales,based on time averages, and associated with higher derivatives, are estimatedin a similar manner. The second task is to address the issue of intermittency :it is shown how the time axis is broken up into very short intervals on whichvarious quantities have lower bounds, larger than long time-averages, which arethemselves interspersed by longer, more quiescent, intervals of time.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for  the special issue on mathematical fluid mechanic

Estimates for the two-dimensional Navier–Stokes equations in terms of the Reynolds number