Isolating constants of motion for the homogeneous turbulence of two and three dimensions

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Abstract
For the inviscid eddy motion in a finite‐dimensioned Fourier space, it is stated that energy and enstrophy are the isolating constants of motion for the 2D homogeneous turbulence. In contrast, the 3D isotropic turbulence has energy as the only constant of motion. If we relax the reflexional invariance, however, helicity emerges as another invariant; hence energy and helicity are said to be the isolating constants of motion for the helical turbulence. Although these are the key assumptions in the construction of equilibrium distributions, they have heretofore been accepted, without proof, as a natural property of the Navier–Stokes dynamics. This paper provides the proof. We have shown here that quadratic constants of motion for the individual triad‐interactions collapse to energy–enstrophy in 2D, but to energy and helicity in 3D.