Further evidence for the persistence of bump-on-tail unstable reduced velocity distribution in the Earth's electron foreshock is presented. This persistence contradicts our understanding of quasi-linear saturation of the bump-on-tail instability; the distributions should be stabilized through velocity space diffusion too quickly to allow an observation of their unstable form. A modified theory for the saturation of the bump-on-tail instability in the Earth's foreshock is proposed and examined using numerical simulation and quasi-linear theoretical techniques. It is argued the mechanism due to Filbert and Kellogg and to Cairns which is responsible for the creation of the bump-on-tail velocity distribution in the foreshock is still operative during the evolution of the bump-on-tail instability. The saturated state of the plasma must represent a balance between this creation mechanism and velocity space diffusion; the saturated state is not determined by velocity space diffusion alone. Thus the velocity distribution of the saturated stat may still appear bump-on-tail unstable to standard linear analysis which does not take the creation mechanism into account. The bump-on-tail velocity distributions in the foreshock would then represent the state of the plasma after saturation of the bump-on-tail instability, not before.

Numerical simulations of two types are discussed. In the first, the particle distribution function and the field are assumed periodic over a spatial invertal of several tens or hundreds of Debye lengths along the local magnetic field. In the second, nonperiodic boundary conditions are imposed on the distribution function to model the spatial gradients which are responsible for the creation of the bump-on-tail unstable distribution. This mechanism for creating the unstable distribution continues to drive the particle distribution unstable during the evolution of the bump-on-tail instability. The saturated velocity distributions of the periodic and nonperiodic simulated plasmas are compared. At saturation the periodic velocity distribution exhibits plateau formation while the nonperiodic distribution does not. A new quasi-linear theory for the space-averaged velocity distribution function is constructed. This theory includes the effects of nonperiodic boundary conditions; it can be used to compare the evolution of periodic and nonperiodic bump-on-tail unstable particle distributions. In addition, this theory is based on the filtered Vlasov-Maxwell system of equations; velocity space filamentation has been removed from the distribution function. It is shown that this theory is applicable on a finite spatial interval within which the field has been expressed in terms of a discrete Fourier mode expansion. Thus direct comparisons between theory and simulation are possible. Some predictions of this new quasi-linear theory are compared favorable to the results of the bump-on-tail simulations.