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In this paper, with the aid of direct numerical simulations (DNS) of forcedturbulence in a periodic domain, we mathematically reformulate theKolmogorov-Richardson energy cascade in terms of vortex stretching. By usingthe description, we prove that if the Navier-Stokes flow satisfies a newregularity criterion in terms of the enstrophy production rate, then the flowdoes not blow up. Our DNS results seem to support this regularity criterion.Next, we mathematically construct the hierarchy of tubular vortices, which isstatistically self-similar in the inertial range. Under the assumptions of thescale-locally of the vortex stretching/compressing (i.e. energy cascade)process and the statistical independence between vortices that are not directlystretched or compressed, we can derive the $-5/3$ power law of the energyspectrum of statistically stationary turbulence without directly using theKolmogorov hypotheses.

Mathematical reformulation of the Kolmogorov-Richardson energy cascade in terms of vortex stretching