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Reduced fluid models including electron inertia and ion finite Larmor radiuscorrections are derived asymptotically, both from fluid basic equations andfrom a gyrofluid model. They apply to collisionless plasmas with smallion-to-electron equilibrium temperature ratio and low $\beta_e$, where$\beta_e$ indicates the ratio between the equilibrium electron pressure and themagnetic pressure exerted by a strong, constant and uniform magnetic guidefield. The consistency between the fluid and gyrofluid approaches is ensuredwhen choosing ion closure relations prescribed by the underlying ordering. Atwo-field reduction of the gyrofluid model valid for arbitrary equilibriumtemperature ratio is also introduced, and is shown to have a noncanonicalHamiltonian structure. This model provides a convenient framework for studyingkinetic Alfv\'en wave turbulence, from MHD to sub-$d_e$ scales (where $d_e$holds for the electron skin depth). Magnetic energy spectra arephenomenologically determined within energy and generalized helicity cascadesin the perpendicular spectral plane. Arguments based on absolute statisticalequilibria are used to predict the direction of the transfers, pointing outthat, within the sub-ion range associated with a $k_\perp^{-7/3}$ transversemagnetic spectrum, the generalized helicity could display an inverse cascade ifinjected at small scales, for example by reconnection processes.

Fluid and gyrofluid modeling of low-$\beta_e$ plasmas: phenomenology of kinetic Alfv\'en wave turbulence