The role of helicity in triad interactions in 3D turbulence investigated in a new shell model

Abstract

Fully developed homogeneous isotropic turbulence in 2D is fundamentally different from 3D. In 2D, the simultaneous conservation of both energy and enstrophy in the inertial ranges of scales leads to a forward cascade of enstrophy and a reverse cascade of energy. In 3D, helicity, the integral of the scalar product of velocity and vorticity, is also an inviscid flow invariant along with kinetic energy. Unlike enstrophy, however, helicity does not block the cascade of energy to small scales. Energy and helicity are not only globally conserved but also conserved within each non-linear triadic interaction between three plane waves in the spectral form of the Navier--Stokes equation (NSE). By decomposing each plane wave into two helical modes of opposite helicities each triadic interaction is split into a set of eight triadic interactions between helical modes (Waleffe 1992). Biferale et al. (2012) recently found that a subset of these interactions which render both signs of helicity separately conserved (i.e. enstrophy-like) leads to an inverse cascade of (part of) the energy. Motivated by this finding we introduce a new shell model obtained from the NSE expressed in the helical basis (Waleffe 1992). By analysing and integrating the new model we attempt to explain why the dual forward cascade of energy and helicity dominates in 3D turbulence.