On the validity of the quasi-linear approximation for turbulent plasmas

The theoretical conditions for the applicability of the quasi-linear theory (QLT) are formulated and to investigate the importance of these conditions, numerical experiments are performed on two different types of weak one-dimensional plasma instabilities. It is found that the behaviour of the bump-in-tail instability is consistent with QLT (saturation field energy < 2 >/8π nT ∼ linear growth rate γ) if these conditions are satisfied, while in the opposite case trapping stabilization is important (trapped-particle bounce frequency ω b ∼γ). The weak symmetric two-stream instability never satisfies the criterion for QLT validity, since all particles that are supposed to diffuse are trapped (Δv diff ∼ Δv trap ). Incidentally, the first saturation level predicted by QLT is the same as given by the condition for trapping stabilization, ω b ∼ γ. However, the important deviation from the weakly turbulent behaviour is apparent from the coherent mode-coupling process, which becomes (relatively) the stronger, the weaker the instability.