  A two-field Hamiltonian gyrofluid model for kinetic Alfv\'en waves retaining
ion finite Larmor radius corrections and parallel magnetic field fluctuations
is used to study direct turbulent cascades from the MHD to the sub-ion scales.
For moderate energy imbalance and weak enough magnetic fluctuations, the
spectrum of the transverse magnetic field and that of the most energetic wave
display a steep transition zone near the ion scale, while the parallel transfer
(and thus the parallel dissipation) remains weak. In this regime, the
perpendicular flux of generalized cross helicity displays a significant decay
past the ion scale, while the perpendicular energy flux remains almost
constant. A phenomenological model suggests that the interactions between
co-propagative waves present at the sub-ion scales can play a central role in
the development of a transition zone in the presence of a helicity barrier.
