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The 1D bump-on-tail problem is studied in order to determine the influence ofdrag on quasi-steady solutions near marginal stability ($1-\gamma_d/\gamma_L\ll1$) when effective collisions are much larger than the instability growth rate($\nu \gg \gamma$). In this common tokamak regime, it is rigorously shown thatthe paradigmatic Berk-Breizman cubic equation for the nonlinear mode evolutionreduces to a much simpler differential equation, dubbed the time-local cubicequation, which can be solved directly. It is found that in addition toincreasing the saturation amplitude, drag introduces a shift in the apparentoscillation frequency by modulating the saturated wave envelope. Excellentagreement is found between the analytic solution for the mode evolution andboth the numerically integrated Berk-Breizman cubic equation and fullynonlinear 1D Vlasov simulations. Experimentally isolating the contribution ofdrag to the saturated mode amplitude for verification purposes is explored butcomplicated by the reality that the amount of drag can not be variedindependently of other key parameters in realistic scenarios. While the effectof drag is modest when the ratio of drag to scattering $\alpha/\nu$ is verysmall, it can become substantial when $\alpha/\nu \gtrsim 0.5$, suggesting thatdrag should be accounted for in quantitative models of fast-ion-driveninstabilities in fusion plasmas.

Analytic quasi-steady evolution of a marginally unstable wave in the presence of drag and scattering