Nonlinear saturation and oscillations of collisionless zonal flows

Abstract

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.