Alfvén waves of arbitrary amplitude with constant total pressure are known to provide an exact solution to the compressible magnetohydrodynamic (MHD) system in a homogeneous plasma, in that nonlinearities are turned off and there no couplings with compressible modes. However, such a dynamical system is linearly unstable to parametric instabilities and large-amplitude Alfvén waves are known to decay into compressible and secondary Alfvénic modes through three- or four-wave resonances that lead to a variety of parametric instabilities, depending on the plasma beta and dispersive effects, such is the case of parametric decay (Galeev & Oraevskii 1973; Derby 1978), modulational, and beat instabilities (Sakai & Sonnerup 1983; Wong & Goldstein 1986; Jayanti & Hollweg 1993; Nariyuki & Hada 2007). Parametric instabilities of Alfvén waves (or of a spectrum of Alfvén waves) have been widely studied over the years through theoretical approaches (Goldstein 1978; Jayanti & Hollweg 1993; Malara & Velli 1996), and numerical simulations adopting both MHD (Ghosh et al. 1994; Malara et al. 2000; Del Zanna et al. 2001; Tenerani et al. 2017) and kinetic models (Terasawa et al. 1986; Matteini et al. 2010; Nariyuki et al. 2012; Verscharen et al. 2012), although most often in one-dimensional setups. In particular, the traditional parametric decay instability has attracted much attention over the years in the context of both turbulence and plasma heating. This type of decay is most efficient at low values of the plasma beta and it essentially involves the decay of a pump Alfvén wave into a lower-frequency reflected Alfvén wave and a forward sound wave. For this reason, parametric decay remains an appealing process because it provides a natural mechanism for the production of reflected modes, which is essential for the triggering of a turbulent cascade. Indeed, recently it has been proposed as a viable mechanism to initiate the turbulent cascade in the solar wind acceleration region (Chandran 2018; Réville et al. 2018), while global MHD simulations of the solar wind have also shown that the parametric decay instability can contribute substantially to solar wind heating and acceleration, thanks to the generation of compressible modes that, in the absence of kinetic effects, naturally steepen into shocks (see, e.g., Shoda et al. 2019). The traditional parametric decay has been also invoked as a possible source for the generation of inward modes and solar wind turbulence in the inner heliosphere, where an increasing content of reflected waves (cross-helicity) and an evolving turbulent spectrum are observed with increasing heliocentric distance (Bavassano et al. 2000). However, expansion effects are known to inhibit its development, essentially because the parametric decay process is strongly suppressed as the plasma beta increases at larger heliocentric distances (Tenerani & Velli 2013, 2020; Del Zanna et al. 2015). Temperature anisotropies can destabilize the parametric decay at values of the plasma beta approaching unity and above, but the anisotropy in the solar wind is not large enough to affect the instability significantly (Tenerani et al. 2017).
Citation: Nariyuki & Hada 2007
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