The triad-interaction representation of homogeneous turbulence

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Abstract
The triad‐interaction representation has been presented for the 2D and 3D homogeneous flows. This has several advantages over the usual Fourier‐amplitude representation: (i) The incompressibility is built into the equation as in the vorticity equation. (ii) For a given wave vector, the number of dynamic equations is one less than that of the Fourier‐amplitude equations. (iii) In the inviscid limit, energy and enstrophy are conserved in 2D, whereas the 3D flow conserves energy and helicity. (iv) The entire family of triad interactions is categorized into two classes in 2D and four classes in 3D, according to the geometry of triad wave vectors. Lastly, (v) the necessary conditions for isotropy in 3D emerge as the reflexional, rotational, and spherical symmetries in the wave vector space, whereas polar symmetry is only the requirement in 2D. The triad‐interaction representation has proved very useful in the investigation of isolating constants of motion and the statistical theory of nonisotropic turbulence.
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