The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L^2m-norms of vorticity D_m (1 ≤ m < ∞). The first in this hierarchy, D_1, is the global enstrophy. Three regimes naturally occur in the D_1-D_m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime DGGKPV13 ; new

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... rical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.