  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.
