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 $N\sim {\gamma }_{\mathrm{MI}}/{\omega }_{\mathrm{DW}}$, where ${\gamma }_{\mathrm{MI}}$ is the MI growth rate and ${\omega }_{\mathrm{DW}}$ is the linear DW frequency. We argue that at $N\ll 1$, 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 $N\gtrsim 1$, 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 $\text{et al}$. [J. Fluid Mech. $\text{654}$, 207 (2010)] and Gallagher $\text{et al}$. [Phys. Plasmas $\text{19}$, 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.

arXiv commentThe associated dataset is available at http://doi.org/10.5281/zenodo.2563449

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Cited by in corpus (6)

• Theory of the tertiary instability and the Dimits shift from reduced drift-wave models
• Zonally dominated dynamics and Dimits threshold in curvature-driven ITG turbulence
• Wave-kinetic approach to zonal-flow dynamics: recent advances
• Formation of solitary zonal structures via the modulational instability of drift waves
• Nonlinear saturation and oscillations of collisionless zonal flows
• Structure formation in turbulence as instability of effective quantum plasma