Transfer of energy from large to small scales in turbulent flows is described as a flux of energy from small wave numbers to large wave numbers in the spectral representation of the Navier–Stokes equation (1.17). The problem of resolving the relevant scales in the flow corresponds in the spectral representation to determining the spectral truncation at large wave numbers. The effective number of degrees of freedom in the flow depends on the Reynolds number. The Kolmogorov scale η depends on Reynolds number as η ∼ Re−¾ (1.11), so the number of waves N necessary to resolve scales larger than η grows with Re as N ∼ η−3 ∼ Re9/4. This means that even for moderate Reynolds numbers ∼ 1000, the effective number of degrees of freedom is of the order of 107. A numerical simulation of the Navier–Stokes equation for high Reynolds numbers is therefore impractical without some sort of reduction of the number of degrees of freedom. Such a calculation with a reduced set of waves was first carried out by Lorenz (1972) in the case of the vorticity equation for 2D turbulence.

The idea is to divide the spectral space into concentric spheres, see Figure 3.1. The spheres may be given exponentially growing radii kn = λn, where λ > 1 is a constant. The set of wave numbers contained in the nth sphere not contained in the (n − 1)th sphere is called the nth shell.