Introduction

In principle, the idea that solutions of the Navier–Stokes equations (NSE) might be adequately represented in a finite-dimensional space arose as a result of the realization that the rapidly varying, high-wavenumber components of the turbulent flow decay so rapidly as to leave the energy-carrying (lower-wavenumber) modes unaffected. With the understanding gained from Kolmogorov's [1941a,b] phenomenological theory (see also Section 3), it appeared that, in 3-dimensional turbulent flows, only wavenumbers up to the cutoff value κd = (∈/ν3)1/4 need be considered. This is the boundary between the inertial range, which is dominated by the inertial term in the equation, and the dissipation range, which is dominated by the viscous term. As explained by Landau and Lifshitz [1971], the question is then reduced to finding the number of resolution elements needed to describe the velocity field in a volume – say, a cube of length ℓ0 on each side. Clearly, if the smallest resolved distance is to be ℓd = 1/κd, then the number of resolution elements is simply (ℓ0/ℓd)3. On adducing some phenomenological and intuitive arguments, it was argued that this ratio is Re9/4, where Re is the Reynolds number. An alternate way to count the number of active modes is as follows: since these modes are those in the inertial range, their frequency κ satisfies κ0 < κ < κd, with κ0 = 1/ℓ0; we conclude that, for κd/κ0 large, that number is of the order of (κd/κ0)3 = (ℓ0/ℓd)3.