Line broadened quasi-linear burst model

A quasi-linear model is developed to produce realistic self-consistent saturation levels when modes do not overlap, and give self-consistent diffusion and wave evolution when modes do overlap. Both regimes give steady or pulsating behaviour in weakly driven systems with classical relaxation and background dissipation present. An avalanche response is demonstrated: wave momentum release caused by the overlap of closely spaced modes can produce mode overlap of more widely spaced modes (a domino effect) or the growth of modes which would be stable in systems unaffected by the diffusion of closely spaced modes. Detailed analysis and calculations are performed for the bump-on-tail instability, and extension of the method to more general problems is briefly discussed.