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

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number Gr , whose character depends on the ratio of the forcing to the viscosity ν, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number Re, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias [23] to the two-dimensional Navier-Stokes equations on a periodic domain [0, L] by estimating quantities of physical relevance, particularly long-time averages 〈·〉, in terms of the Reynolds number Re = Ul/ν, where U = L 〈 ‖u‖ 2 〉 and l is the forcing scale. In particular, the Constantin-FoiasTemam upper bound [1] on the attractor dimension converts to a2lRe (1 + lnRe) 1/3 , while the estimate for the inverse Kraichnan length is (a2lRe) , where al is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in 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 which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.