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Title: 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; yet, 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 ~ γ MIDW, where γ MI is the MI growth rate and ω 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 ≳ 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.

Authors:
ORCiD logo; ORCiD logo; ORCiD logo
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
OSTI Identifier:
1525338
Alternate Identifier(s):
OSTI ID: 1544248
Grant/Contract Number:  
[AC02-09CH11466]
Resource Type:
Published Article
Journal Name:
New Journal of Physics
Additional Journal Information:
[Journal Name: New Journal of Physics Journal Volume: 21 Journal Issue: 6]; Journal ID: ISSN 1367-2630
Publisher:
IOP Publishing
Country of Publication:
United Kingdom
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; collisionless zonal flows; modulational instability; nonlinear stage; predator-prey oscillations

Citation Formats

Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. Nonlinear saturation and oscillations of collisionless zonal flows. United Kingdom: N. p., 2019. Web. doi:10.1088/1367-2630/ab2251.
Zhu, Hongxuan, Zhou, Yao, & Dodin, I. Y. Nonlinear saturation and oscillations of collisionless zonal flows. United Kingdom. doi:10.1088/1367-2630/ab2251.
Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. Sat . "Nonlinear saturation and oscillations of collisionless zonal flows". United Kingdom. doi:10.1088/1367-2630/ab2251.
@article{osti_1525338,
title = {Nonlinear saturation and oscillations of collisionless zonal flows},
author = {Zhu, Hongxuan and Zhou, Yao and Dodin, I. Y.},
abstractNote = {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; yet, 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 ~ γ MI/ω DW, where γ MI is the MI growth rate and ω 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 ≳ 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.},
doi = {10.1088/1367-2630/ab2251},
journal = {New Journal of Physics},
number = [6],
volume = [21],
place = {United Kingdom},
year = {2019},
month = {6}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1088/1367-2630/ab2251

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Cited by: 2 works
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Figures / Tables:

Figure 1 Figure 1: (a) The effective potential $Θ$($\bar{u}$) [see (3.8)] with $\bar{a}$0 = 4 (and hence g2 = 1 > 0). The MI corresponds to a initial condition $\bar{u}$ = $\bar{u}$0 near the origin (black cross, $\bar{u}$0 = 10-3). At the nonlinear stage, $\bar{u}$ is constrained by the conservation of themore » `energy', E $\dot{=}$ (d$τ$$\bar{u}$)2=2+$Θ$ (dashed line), and hence will start to decrease when reaching $\bar{u}$ ≈ 2 (black circle). (b) Numerical solutions of the 4MT system governed by (3.4) and (3.5) with dimensionless variables given by (3.6). The coefficient $v$ describes an ad hoc damping that mimics the coupling to other DW sidebands beyond the 4MT [see (A.11)]. The initial conditions are $\bar{a}$ = $\bar{a}$0 = 4, $\bar{u}$ = 10-3, and $\bar{b}$ = $\bar{c}$ = 0. A transition from oscillations to saturation of $\bar{u}$ is observed as $v$ increases.« less

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