Quasi-linear theory of shear Alfvén waves driven by a bump-on-tail distribution

A magnetized plasma in a shearless slab geometry is considered. The plasma ions have a distribution, in velocity space, of the bump-on-tail type. The plasma is assumed to be spatially homogeneous. Shear Alfvén waves can be destabilized via wave-ion resonances near the bump. A quasi-linear theory describing the saturation of the unstable wave is developed for the case where a single mode description is appropriate.