Leith

A Hamilitonian 2-field reduced gyrofluid model for kinetic Alfvén waves taking into account ion FLR corrections, parallel magnetic field fluctuations and electron inertia, is used to study turbulent cascades, from the MHD to the electron ranges, in the case of imbalance between waves propagating along or opposite to the direction of the ambient magnetic field. The weak turbulence formalism in the absence of electron inertia leads to kinetic equations for the spectral densities of total energy and generalized cross-helicity, which reduce to those of RMHD at large scales, and REMHD at small scales. Leith-type nonlinear diffusion equations are derived in the limit of ultra-local interactions and a phenomenological formulation is obtained for the strong turbulence regime. These equations are studied analytically and integrated numerically. For a given level of imbalance in the MHD range, the flux of cross-helicity is much smaller when a dispersive range is present before dissipation scales are reached. Large imbalance leads to steeper sub-ion range spectra.