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Large amplitude Alfv\'en waves tend to be unstable to parametricinstabilities which result in a decay process of the initial wave intodifferent daughter waves depending upon the amplitude of the fluctuations andthe plasma beta. The propagation angle with respect to the mean magnetic fieldof 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 hybridsimulations. In particular, we study the decay and the subsequent nonlinearevolution of large-amplitude Alfv\'en waves by investigating the saturationmechanism of the instability and its final nonlinear state reached fordifferent wave amplitudes and plasma beta conditions. As opposed toone-dimensional simulations where the Decay instability is suppressed forincreasing plasma beta values, we find that the decay process inmulti-dimensions persists at large values of the plasma beta via thefilamentation/magnetosonic decay instabilities. In general, the decay processacts as a trigger both to develop a perpendicular turbulent cascade and toenhance mean field-aligned wave-particle interactions. We find indeed that thesaturated state is characterized by a turbulent plasma displaying afield-aligned beam at the Alfv\'en speed and increased temperatures that weascribe to the Landau resonance and pitch angle scattering in phase space.

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