  A two-field Hamiltonian gyrofluid model for kinetic Alfv\'en waves retaining
ion finite Larmor radius corrections, parallel magnetic field fluctuations and
electron inertia, is used to study turbulent cascades from the MHD to the
sub-ion scales. Special attention is paid to the case of imbalance between
waves propagating along or opposite to the ambient magnetic field. For weak
turbulence in the absence of electron inertia, kinetic equations for the
spectral density of the conserved quantities (total energy and generalized
cross-helicity) are obtained. They provide a global description, matching
between the regimes of reduced MHD at large scales and electron reduced MHD at
small scales, previously considered in the literature. In the limit of
ultra-local interactions, Leith-type nonlinear diffusion equations in the
Fourier space are derived and heuristically extended to the strong turbulence
regime by modifying the transfer time appropriately. Relations with existing
phenomenological models for imbalanced MHD and balanced sub-ion turbulence are
discussed. It turns out that in the presence of dispersive effects, the
dynamics is sensitive on the way turbulence is maintained in a steady state.
Furthermore, the total energy spectrum at sub-ion scales becomes steeper as the
generalized cross-helicity flux is increased.
