super-level sets of the first-order spatial fluctuation of the velocity. It turns out that considering the super-level sets of the higher-order spatial fluctuations of the velocity yields a further reduction of the scaling gap, and the ultimate goal of showing asymptotic criticality as the order of the spatial fluctuation goes to infinity is in reach [@GrXu19].

Appendix: Homogeneous, isotropic turbulence
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Here, we consider a forced isotropic turbulence simulation made available through the JHTDB. The flow is kept in the statistical stationary state of the fully developed turbulence by perpetually injecting the energy at large scales (at low Fourier modes, effectively), keeping the kinetic energy constant within the Fourier modes of the length less or equal to two. Any stationarity is exhibited in the average only, and any pointwise-in-time-measured extremal quantities (e.g., the vorticity maximum, $d$ and $r$) are expected to exhibit significant fluctuations.

As already noted in Introduction, the scale of sparseness is a small scale and–as such–is expected to trend as a dissipation range quantity. In particular, in the framework of the $Z_\alpha$-classes, we expect it to stay within the confinements of the critical diffusion/dissipation class $Z_\frac{1}{2}$. This was indeed confirmed via time series analysis of the data 