[@BdVBe02] where it was shown that the assumption on the Lipschitz-continuity can be scaled down to $\frac{1}{2}$-Hölder-continuity, [@Gr09] in which a complete *spatiotemporal localization* of the $\frac{1}{2}$-Hölder-continuity regularity criterion was performed, and [@GrGu10] in which the authors presented a two-parameter family of local, hybrid geometric-analytic, *scaling-invariant* regularity criteria, based on a *dynamic balance* between the coherence of the vorticity direction and the vorticity magnitude.

Another (in addition to the local coherence of the vorticity direction) morphological signature of the regions of intense vorticity is *spatial intermittency*–this has been well-documented in computational simulations as well as experiments. Classical references on the morphology of turbulent flows include [@AKKG87; @JWSR93; @S81; @SJO91; @VM94]. One should also note that–in contrast to the more traditional approach of chasing the super-high Reynolds number regimes–a relatively recent work [@SchSree] revealed that, in sufficiently resolved flows, the universality in the realm of the small-scale structures already transpires at low to moderate Reynolds numbers.

The concepts of local 1D and 3D ‘sparseness’ introduced in [@Gr13] were designed to model the spatial intermittency in a way amenable to rigorous mathematical analysis based on the 3D NSE. Henceforth, sparseness will refer to 3D sparseness defined below ($m^3$ denotes the 3-dimensional Lebesgue 