The 1D bump-on-tail problem is studied in order to determine the influence of drag on quasi-steady solutions near marginal stability ($1-{\gamma }_d/{\gamma }_L\ll 1$) when effective collisions are much larger than the instability growth rate ($\nu \gg \gamma$). In this common tokamak regime, it is rigorously shown that the paradigmatic Berk–Breizman cubic equation for the nonlinear mode evolution reduces to a much simpler differential equation, dubbed the time-local cubic equation, which can be solved directly. It is found that in addition to increasing the saturation amplitude, drag introduces a shift in the apparent oscillation frequency by modulating the saturated wave envelope. Excellent agreement is found between the analytic solution for the mode evolution and both the numerically integrated Berk–Breizman cubic equation and fully nonlinear 1D Vlasov simulations. Experimentally isolating the contribution of drag to the saturated mode amplitude for verification purposes is explored but complicated by the reality that the amount of drag cannot be varied independently of other key parameters in realistic scenarios. While the influence of drag is modest when the ratio of effective drag to effective scattering $\alpha /\nu$ is very small, it can become substantial when $\alpha /\nu \gtrsim 0.5$, suggesting that drag should be accounted for in quantitative models of fast-ion-driven instabilities in fusion plasmas.

中文翻译：

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存在阻力和散射的边缘不稳定动力学等离子体不稳定性的解析准稳态演化

研究一维碰撞尾部问题以确定阻力对边缘稳定性附近的准稳态解的影响（$1-{\gamma }_d/{\gamma }_{升}\ll 1$) 当有效碰撞远大于不稳定性增长率 ($\nu \gg \gamma$）。在这种常见的托卡马克机制中，严格证明了非线性模式演化的范式 Berk-Breizman 三次方程简化为更简单的微分方程，称为时域三次方程，可以直接求解。发现除了增加饱和幅度之外，阻力通过调制饱和波包络引入了表观振荡频率的偏移。在模式演化的解析解与数值积分的 Berk-Breizman 三次方程和​​完全非线性 1D Vlasov 模拟之间发现了极好的一致性。出于验证目的，通过实验隔离阻力对饱和模式振幅的贡献进行了探索，但由于阻力量不能独立于现实场景中的其他关键参数而变化的现实而变得复杂。而当有效阻力与有效散射的比值时，阻力的影响不大$\alpha /\nu$ 非常小，当它可以变得很大时 $\alpha /\nu \gtrsim 0.5$，表明在聚变等离子体中快离子驱动不稳定性的定量模型中应考虑阻力。