Kinetic Alfv\'en waves generation by large-scale phase-mixing

One view of the solar-wind turbulence is that the observed highly anisotropic fluctuations at spatial scales near the proton inertial length $d_p$ may be considered as Kinetic Alfv\'en waves (KAWs). In the present paper, we show how phase-mixing of large-scale parallel propagating Alfv\'en waves is an efficient mechanism for the production of KAWs at wavelengths close to $d_p$ and at large propagation angle with respect to the magnetic field. Magnetohydrodynamic (MHD), Hall-Magnetohydrodynamic (HMHD), and hybrid Vlasov-Maxwell (HVM) simulations modeling the propagation of Alfv\'en waves in inhomogeneous plasmas are performed. In linear regime, the role of dispersive effects is singled out by comparing MHD and HMHD results. Fluctuations produced by phase-mixing are identified as KAWs through a comparison of polarization of magnetic fluctuations and wave group velocity with analytical linear predictions. In the nonlinear regime, comparison of HMHD and HVM simulations allows to point out the role of kinetic effects in shaping the proton distribution function. We observe generation of temperature anisotropy with respect to the local magnetic field and production of field-aligned beams. The regions where the proton distribution function highly departs from thermal equilibrium are located inside the shear layers, where the KAWs are excited, this suggesting that the distortions of the proton distribution are driven by a resonant interaction of protons with KAW fluctuations. Our results are relevant in configurations where magnetic field inhomogeneities are present, as, for example, in the solar corona where the presence of Alfv\'en waves has been ascertained.