The nature of triad interactions in homogeneous turbulence

Abstract

Nonlinear interactions in homogeneous turbulence are investigated using a decomposition of the velocity field in terms of helical modes. There are two helical modes per wave vector and thus eight fundamental triad interactions. These eight elementary interactions fit in only two classes, ‘‘R’’ (for ‘‘reverse’’) and ‘‘F’’ (for ‘‘forward’’), depending on whether the small-scale helical modes have helicities of the same or of the opposite sign. In a single-triad interaction, the large scale is unstable when the small-scale helical modes have helicities of opposite signs (class ‘‘F’’), and the medium scale is unstable otherwise (class ‘‘R’’). It is proposed that, on average, the triple correlations in a turbulent flow correspond to these unstable states. In the limit of nonlocal triads, where one leg is much smaller than the other two, the triadic interscale energy transfer is largest for interactions of class ‘‘R.’’ In that case, most of the energy flows locally in wave number, from the medium scale to the smallest, with a comparatively small feedback on the large scale. However, integrating over all scales in an inertial range, the net effect of nonlocal interactions of class ‘‘R’’ is a reverse energy cascade from small to large scales. All other interactions transfer energy upward in wave number. In local triads, this upward energy transfer occurs primarily between modes with helicities of the same sign, through catalytic interactions with a mode whose helicity has the opposite sign. The class ‘‘F’’ interactions transfer energy to the small scales and exist only in three dimensions. The physical processes associated to both classes of interactions are discussed. It is shown that the large local transfers due to nonlocal ‘‘R’’ interactions appear in pairs of opposite signs that nearly cancel each other and the net effect corresponds to an advection in wave space.