Parametric instabilities have often been invoked to explain some of the features of Alfvénic turbulence as observed in the (fast) solar wind plasma, namely the gradual reduction of cross helicity and its final saturation with heliocentric distance. Moreover, this instability could also be responsible for direct plasma heating and creation of transverse small-scale Alfvénic fluctuations, as required by recent models for coronal heating and solar wind acceleration. Here we discuss numerical simulations of the long-term nonlinear evolution of Alfvén waves, both in the MHD and hybrid regimes, performed by the plasma theory group in Florence in the last decade and we show, for the first time, 2-D MHD simulations of the parametric decay of oblique arc-polarized waves.

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