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Fully developed homogeneous isotropic turbulence in 2D is fundamentallydifferent from 3D. In 2D, the simultaneous conservation of both energy andenstrophy in the inertial ranges of scales leads to a forward cascade ofenstrophy and a reverse cascade of energy. In 3D, helicity, the integral of thescalar product of velocity and vorticity, is also an inviscid flow invariantalong with kinetic energy. Unlike enstrophy, however, helicity does not blockthe cascade of energy to small scales. Energy and helicity are not onlyglobally conserved but also conserved within each non-linear triadicinteraction between three plane waves in the spectral form of theNavier--Stokes equation (NSE). By decomposing each plane wave into two helicalmodes of opposite helicities each triadic interaction is split into a set ofeight triadic interactions between helical modes (Waleffe 1992). Biferale etal. (2012) recently found that a subset of these interactions which render bothsigns of helicity separately conserved (i.e. enstrophy-like) leads to aninverse cascade of (part of) the energy. Motivated by this finding we introducea new shell model obtained from the NSE expressed in the helical basis (Waleffe1992). By analysing and integrating the new model we attempt to explain why thedual forward cascade of energy and helicity dominates in 3D turbulence.

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