Dynamical behavior of the fundamental triad-interaction system in three-dimensional homogeneous turbulence

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Abstract
The fundamental triad‐interaction system is the building block of the nonlinear interaction of the three‐dimensional Navier–Stokes equations. It satisfies incompressibility, conserves energy and helicity, preserves measure under the time evolution, and involves an arbitrary parameter taking any value in [0,2π]. Under several values of this parameter, it is found that the fundamental triad‐interaction system cannot be ergodic on the constant energy‐helicity surface because of the extraneous constants of motion other than energy and helicity. This at once implies that isotropy is not possible for the spectral tensor, for isotropy calls for mixing which is a stronger requirement than ergodicity. Although the dynamics of a single triad‐interaction system is not directly relevant to real homogeneous turbulence, a theoretical framework to investigate ergodicity and mixing and isotropy of the three‐dimensional turbulence model by consistently including many fundamental triad‐interaction systems is provided.