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The accuracy of quasilinear theory applied to the electron bump-on-tailinstability, a classic model problem, is explored with conservative high-orderdiscontinuous Galerkin methods applied to both the quasilinear equations and toa direct simulation of the Vlasov-Poisson equations. The initial condition ischosen in the regime of beam parameters for which quasilinear theory should beapplicable. Quasilinear diffusion is initially in good agreement with thedirect simulation but later underestimates the turbulent momentum flux. Thedirect simulation corrects from quasilinear evolution by quenching theinstability in a finite time and producing a robust state of oscillation. Fluxenhancement above quasilinear levels occurs as the phase space eddy turnovertime in the largest amplitude wavepackets becomes comparable to the transittime of resonant phase fluid through wavepacket potentials. In this regimeeddies effectively turn over during wavepacket transit so that phase fluidpredominantly disperses by eddy phase mixing rather than by randomly phasedwaves. The enhanced turbulent flux of resonant phase fluid leads in turn,through energy conservation, to an increase in non-resonant turbulent flux andthus to an enhanced heating of the main thermal body above quasilinearpredictions. These findings shed light on the kinetic turbulence fluctuationspectrum and support the theory that collisionless momentum diffusion beyondthe quasilinear approximation can be understood through the dynamics of phasespace eddies (or clumps and granulations).

On the validity of quasilinear theory applied to the electron bump-on-tail instability