Propagation of non-WKB Alfven waves in a multicomponent solar wind with differential ion flow

The propagation of dissipationless, hydromagnetic, non-WKB, purely toroidal Alfv\'en waves in a realistic background three-fluid solar wind with axial symmetry and differential proton-alpha flow is investigated. The wave equations are derived from standard multi-fluid 5-moment equations. The Alfv\'enic point, where the combined poloidal Alfv\'en Mach number $M_T=1$, is found to be a singular point for the wave equation, which is then numerically solved for three representative angular frequencies $\omega=10^{-3}$, $10^{-4}$ and $10^{-5}$ rad s$^{-1}$ with a fixed wave amplitude of 10 km s$^{-1}$ imposed at the coronal base (1 $R_\odot$). Between 1 $R_\odot$ and 1 AU, the numerical solutions show substantial deviation from the WKB expectations. Even for the relatively high frequency $\omega=10^{-3}$ rad s$^{-1}$, a WKB-like behavior can be seen only in regions $r\gtrsim 10$ $R_\odot$. In the low-frequency case $\omega=10^{-5}$ rad s$^{-1}$, the computed profiles of wave-related parameters show a spatial dependence distinct from the WKB one, the deviation being particularly pronounced in interplanetary space. In the inner corona $r\lesssim 4$ $R_\odot$, the computed ion velocity fluctuations are considerably smaller than the WKB expectations in all cases, as is the computed wave-induced acceleration exerted on protons or alpha particles. With the chosen base wave amplitude, the wave acceleration has negligible effect on the ion force balance in the corona. Hence processes other than the non-WKB wave acceleration are needed to accelerate the ions out of the gravitational potential well of the Sun. However, at large distances beyond the Alfv\'enic point, the low-frequency waves can play an important role in the ion dynamics, with the net effect being to equalize the speeds of the two ion species considered.