  The accuracy of quasilinear theory applied to the electron bump-on-tail
instability, a classic model problem, is explored with conservative high-order
discontinuous Galerkin methods applied to both the quasilinear equations and to
a direct simulation of the Vlasov-Poisson equations. The initial condition is
chosen in the regime of beam parameters for which quasilinear theory should be
applicable. Quasilinear diffusion is initially in good agreement with the
direct simulation but later underestimates the turbulent momentum flux. The
direct simulation corrects from quasilinear evolution by quenching the
instability in a finite time and producing a robust state of oscillation. Flux
enhancement above quasilinear levels occurs as the phase space eddy turnover
time in the largest amplitude wavepackets becomes comparable to the transit
time of resonant phase fluid through wavepacket potentials. In this regime
eddies effectively turn over during wavepacket transit so that phase fluid
predominantly disperses by eddy phase mixing rather than by randomly phased
waves. The enhanced turbulent flux of resonant phase fluid leads in turn,
through energy conservation, to an increase in non-resonant turbulent flux and
thus to an enhanced heating of the main thermal body above quasilinear
predictions. These findings shed light on the kinetic turbulence fluctuation
spectrum and support the theory that collisionless momentum diffusion beyond
the quasilinear approximation can be understood through the dynamics of phase
space eddies (or clumps and granulations).
