The role of parametric instabilities in turbulence generation and proton heating: Hybrid simulations of parallel propagating Alfv\'en waves

Large amplitude Alfv\'en waves tend to be unstable to parametric instabilities which result in a decay process of the initial wave into different daughter waves depending upon the amplitude of the fluctuations and the plasma beta. The propagation angle with respect to the mean magnetic field of the daughter waves plays an important role in determining the type of decay. In this paper, we revisit this problem by means of multi-dimensional hybrid simulations. In particular, we study the decay and the subsequent nonlinear evolution of large-amplitude Alfv\'en waves by investigating the saturation mechanism of the instability and its final nonlinear state reached for different wave amplitudes and plasma beta conditions. As opposed to one-dimensional simulations where the Decay instability is suppressed for increasing plasma beta values, we find that the decay process in multi-dimensions persists at large values of the plasma beta via the filamentation/magnetosonic decay instabilities. In general, the decay process acts as a trigger both to develop a perpendicular turbulent cascade and to enhance mean field-aligned wave-particle interactions. We find indeed that the saturated state is characterized by a turbulent plasma displaying a field-aligned beam at the Alfv\'en speed and increased temperatures that we ascribe to the Landau resonance and pitch angle scattering in phase space.