“…The maximum growth rate γ max = k max V a,0 is obtained at a wave number k max r * = ξ/2M 2 a,0 usually 1 (small scale perturbations). The instability grows much faster than the resonant streaming instability as demonstrated in a series of analytical [21,193,167,195,196,197,198,19] and numerical studies [17,129,199,200,201,26,202,203,204,205] (see §3.4). This process of fast amplification of short-scale modes proposed by [21] can be accompanied with amplification of the long-wavelength fluctuations that would allow the effective confinement and acceleration of higher energy particles [198,203,18,168].…”