  We revisit the applications of quasi-linear theory as a paradigmatic model
for weak plasma turbulence and the associated bump-on-tail problem. The work,
presented here, is built around the idea that large-amplitude or strongly
shaped beams do not relax through diffusion only and that there exists an
intermediate time scale where the relaxations are convective (ballistic-like).
We cast this novel idea in the rigorous form of a self-consistent nonlinear
dynamical model, which generalizes the classic equations of the quasi-linear
theory to "broad" beams with internal structure. We also present numerical
simulation results of the relaxation of a broad beam of energetic particles in
cold plasma. These generally demonstrate the mixed diffusive-convective
features of supra-thermal particle transport; and essentially depend on
nonlinear wave-particle interactions and phase-space structures. Taking into
account modes of the stable linear spectrum is crucial for the self-consistent
evolution of the distribution function and the fluctuation intensity spectrum.
