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 ωb∼γ). The weak symmetric two-stream instability never satisfies the criterion for QLT validity, since all particles that are supposed to diffuse are trapped (Δvdiff ∼ Δvtrap). Incidentally, the first saturation level predicted by QLT is the same as given by the condition for trapping stabilization, ωb ∼ 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.
First principles based simulations of instabilities and turbulence