Nonlinear saturation and oscillations of collisionless zonal flows

Abstract

In homogeneous drift-wave 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 ∼ γMI/ωDW, where γMI is the MI growth rate and ωDW 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 (2010 J. Fluid Mech. 654 207) and Gallagher et al (2012 Phys. Plasmas 19 122115) and offer a different perspective on what the control parameter actually is that determines the transition from the oscillations to saturation of collisionless ZFs.