  The periodic $3D$ Navier-Stokes equations are analyzed in terms of
dimensionless, scaled, $L^{2m}$-norms of vorticity $D_{m}$ ($1 \leq m <
\infty$). 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 \cite{DGGKPV13}\,;
new numerical 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.
