  Reduced fluid models including electron inertia and ion finite Larmor radius
corrections are derived asymptotically, both from fluid basic equations and
from a gyrofluid model. They apply to collisionless plasmas with small
ion-to-electron equilibrium temperature ratio and low $\beta_e$, where
$\beta_e$ indicates the ratio between the equilibrium electron pressure and the
magnetic pressure exerted by a strong, constant and uniform magnetic guide
field. The consistency between the fluid and gyrofluid approaches is ensured
when choosing ion closure relations prescribed by the underlying ordering. A
two-field reduction of the gyrofluid model valid for arbitrary equilibrium
temperature ratio is also introduced, and is shown to have a noncanonical
Hamiltonian structure. This model provides a convenient framework for studying
kinetic Alfv\'en wave turbulence, from MHD to sub-$d_e$ scales (where $d_e$
holds for the electron skin depth). Magnetic energy spectra are
phenomenologically determined within energy and generalized helicity cascades
in the perpendicular spectral plane. Arguments based on absolute statistical
equilibria are used to predict the direction of the transfers, pointing out
that, within the sub-ion range associated with a $k_\perp^{-7/3}$ transverse
magnetic spectrum, the generalized helicity could display an inverse cascade if
injected at small scales, for example by reconnection processes.
