Nonlinear saturation and oscillations of collisionless zonal flows

In homogeneous drift-wave (DW) turbulence, zonal flows (ZFs) can be generated via a modulational instability (MI) that either saturates monotonically or leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is often attributed as the predator-prey oscillations induced by ZF collisional damping; however, similar dynamics is also observed in collisionless ZFs, in which case a different mechanism must be involved. Here, we propose a semi-analytic theory that explains the transition between the oscillations and saturation of collisionless ZFs within the quasilinear Hasegawa-Mima model. By analyzing phase-space trajectories of DW quanta (driftons) within the geometrical-optics (GO) approximation, we argue that the parameter that controls this transition is N ~ \gamma_MI/\omega_DW, where \gamma_MI is the MI growth rate and \omega_DW is the linear DW frequency. We argue that at N << 1, ZFs oscillate due to the presence of so-called passing drifton trajectories, and we derive an approximate formula for the ZF amplitude as a function of time in this regime. We also show that at N >~ 1, the passing trajectories vanish and ZFs saturate monotonically, which can be attributed to phase mixing of higher-order sidebands. A modification of N that accounts for effects beyond the GO limit is also proposed. These analytic results are tested against both quasilinear and fully-nonlinear simulations. They also explain the earlier numerical results by Connaughton et al. [J. Fluid Mech. 654, 207 (2010)] and Gallagher et al. [Phys. Plasmas 19, 122115 (2012)] and offer a revised perspective on what the control parameter is that determines the transition from the oscillations to saturation of collisionless ZFs.

在均匀漂移波（drift-wave, DW）湍流中，带状流（zonal flows, ZFs）可通过调制不稳定性（modulational instability, MI）产生：该不稳定性要么呈现单调饱和行为，要么在非线性阶段引发带状流能量的振荡。这类动力学过程常被归因于由带状流碰撞阻尼诱导的捕食-被捕食振荡；然而在无碰撞带状流中也观测到了类似的动力学行为，此时必然存在不同的作用机制。
本文基于准线性长谷川-三轮模型（quasilinear Hasegawa-Mima model），提出了一套半解析理论，用以阐释无碰撞带状流的振荡与饱和之间的转变过程。通过在几何光学（geometrical-optics, GO）近似框架下分析漂移波量子（drifton，漂移子）的相空间轨迹，本文提出调控该转变的特征参数为$N sim gamma_{ ext{MI}}/omega_{ ext{DW}}$，其中$gamma_{ ext{MI}}$为调制不稳定性的增长率，$omega_{ ext{DW}}$为线性漂移波的频率。本文认为，当$N ll 1$时，由于存在所谓的通行漂移子轨迹，带状流会呈现振荡行为；同时本文推导了该参数区间内带状流振幅随时间变化的近似表达式。本文还证明，当$N gtrsim 1$时，通行漂移子轨迹消失，带状流将呈现单调饱和行为，这一现象可归因于高阶边带的相位混合。本文同时提出了一种修正后的特征参数$N$，用以考虑几何光学近似极限之外的物理效应。上述解析结果通过准线性模拟与全非线性模拟得到了验证。该结果同时解释了Connaughton等人[J. Fluid Mech. 654, 207 (2010)]与Gallagher等人[Phys. Plasmas 19, 122115 (2012)]的早期数值研究结果，并为判定无碰撞带状流从振荡到饱和的转变控制参数提供了全新的视角。