# 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 _{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.

- Authors:

- 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}

}

DOI: 10.1088/1367-2630/ab2251

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

_{0}= 4 (and hence

*g*

^{2}= 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 »

Works referenced in this record:

##
Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations

journal, September 1998

- Lin, Z.
- Science, Vol. 281, Issue 5384

##
Zonal flows in plasma—a review

journal, April 2005

- Diamond, P. H.; Itoh, S-I; Itoh, K.
- Plasma Physics and Controlled Fusion, Vol. 47, Issue 5

##
Generation and Stability of Zonal Flows in Ion-Temperature-Gradient Mode Turbulence

journal, December 2000

- Rogers, B. N.; Dorland, W.; Kotschenreuther, M.
- Physical Review Letters, Vol. 85, Issue 25

##
Electron temperature gradient driven turbulence

journal, May 2000

- Jenko, F.; Dorland, W.; Kotschenreuther, M.
- Physics of Plasmas, Vol. 7, Issue 5

##
Jovian atmospheric dynamics: an update after *Galileo* and *Cassini*

journal, July 2005

- Vasavada, Ashwin R.; Showman, Adam P.
- Reports on Progress in Physics, Vol. 68, Issue 8

##
Zonal flow generation by parametric instability in magnetized plasmas and geostrophic fluids

journal, May 2000

- Smolyakov, A. I.; Diamond, P. H.; Shevchenko, V. I.
- Physics of Plasmas, Vol. 7, Issue 5

##
Coherent Structure Phenomena in Drift Wave–Zonal Flow Turbulence

journal, January 2000

- Smolyakov, A. I.; Diamond, P. H.; Malkov, M.
- Physical Review Letters, Vol. 84, Issue 3

##
On the relation between secondary and modulational instabilities

journal, April 2007

- Strintzi, D.; Jenko, F.
- Physics of Plasmas, Vol. 14, Issue 4

##
Zonostrophic Instability

journal, May 2012

- Srinivasan, Kaushik; Young, W. R.
- Journal of the Atmospheric Sciences, Vol. 69, Issue 5

##
Zonal flow as pattern formation

journal, October 2013

- Parker, Jeffrey B.; Krommes, John A.
- Physics of Plasmas, Vol. 20, Issue 10

##
Generation of zonal flows through symmetry breaking of statistical homogeneity

journal, March 2014

- Parker, Jeffrey B.; Krommes, John A.
- New Journal of Physics, Vol. 16, Issue 3

##
Zonostrophic instability driven by discrete particle noise

journal, April 2017

- St-Onge, D. A.; Krommes, J. A.
- Physics of Plasmas, Vol. 24, Issue 4

##
Generalized Quasilinear Approximation: Application to Zonal Jets

journal, May 2016

- Marston, J. B.; Chini, G. P.; Tobias, S. M.
- Physical Review Letters, Vol. 116, Issue 21

##
Excitation of zonal flow by drift waves in toroidal plasmas

journal, August 2000

- Chen, Liu; Lin, Zhihong; White, Roscoe
- Physics of Plasmas, Vol. 7, Issue 8

##
Modulational instability of Rossby and drift waves and generation of zonal jets

journal, May 2010

- Connaughton, Colm P.; Nadiga, Balasubramanya T.; Nazarenko, Sergey V.
- Journal of Fluid Mechanics, Vol. 654

##
The modulational instability in the extended Hasegawa-Mima equation with a finite Larmor radius

journal, December 2012

- Gallagher, S.; Hnat, B.; Connaughton, C.
- Physics of Plasmas, Vol. 19, Issue 12

##
Rossby and drift wave turbulence and zonal flows: The Charney–Hasegawa–Mima model and its extensions

journal, December 2015

- Connaughton, Colm; Nazarenko, Sergey; Quinn, Brenda
- Physics Reports, Vol. 604

##
Streamer and zonal flow generation from envelope modulations in drift wave turbulence

journal, September 2001

- Champeaux, S.; Diamond, P. H.
- Physics Letters A, Vol. 288, Issue 3-4

##
Nonlinear damping of zonal flows

journal, August 2016

- Koshkarov, O.; Smolyakov, A. I.; Mendonca, J. T.
- Plasma Physics Reports, Vol. 42, Issue 8

##
Influence of the mean flow on zonal flow generation

journal, December 2008

- Lashkin, Volodymyr M.
- Physics of Plasmas, Vol. 15, Issue 12

##
Streamer saturation: a dynamical systems approach

journal, March 2006

- Jenko, F.
- Physics Letters A, Vol. 351, Issue 6

##
Nonlinear interactions between drift waves and zonal flows

journal, July 2002

- Shukla, P. K.; Stenflo, L.
- The European Physical Journal D - Atomic, Molecular and Optical Physics, Vol. 20, Issue 1

##
Zonal flow and streamer generation in drift turbulence

journal, May 2001

- Manfredi, G.; Roach, C. M.; Dendy, R. O.
- Plasma Physics and Controlled Fusion, Vol. 43, Issue 6

##
The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows

journal, December 2001

- Lashmore-Davies, C. N.; McCarthy, D. R.; Thyagaraja, A.
- Physics of Plasmas, Vol. 8, Issue 12

##
The oscillation between Rossby wave and zonal flow in a barotropic fluid

journal, September 1981

- Mahanti, A. C.
- Archives for Meteorology, Geophysics, and Bioclimatology Series A, Vol. 30, Issue 3

##
The stability of planetary waves on an infinite beta‐plane

journal, January 1974

- Gill, A. E.
- Geophysical Fluid Dynamics, Vol. 6, Issue 1

##
On non-local energy transfer via zonal flow in the Dimits shift

journal, October 2017

- St-Onge, Denis A.
- Journal of Plasma Physics, Vol. 83, Issue 5

##
Magnetic Suppression of Zonal Flows on a Beta Plane

journal, August 2018

- Constantinou, Navid C.; Parker, Jeffrey B.
- The Astrophysical Journal, Vol. 863, Issue 1

##
Structure and Spacing of Jets in Barotropic Turbulence

journal, October 2007

- Farrell, Brian F.; Ioannou, Petros J.
- Journal of the Atmospheric Sciences, Vol. 64, Issue 10

##
S3T Stability of the Homogeneous State of Barotropic Beta-Plane Turbulence

journal, May 2015

- Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J.
- Journal of the Atmospheric Sciences, Vol. 72, Issue 5

##
Zonal-flow dynamics from a phase-space perspective

journal, December 2016

- Ruiz, D. E.; Parker, J. B.; Shi, E. L.
- Physics of Plasmas, Vol. 23, Issue 12

##
Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation

journal, January 2019

- Ruiz, D. E.; Glinsky, M. E.; Dodin, I. Y.
- Journal of Plasma Physics, Vol. 85, Issue 1

##
Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation

journal, May 2018

- Zhu, Hongxuan; Zhou, Yao; Ruiz, D. E.
- Physical Review E, Vol. 97, Issue 5

##
On the structure of the drifton phase space and its relation to the Rayleigh–Kuo criterion of the zonal-flow stability

journal, July 2018

- Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
- Physics of Plasmas, Vol. 25, Issue 7

##
Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations

journal, November 2016

- Parker, Jeffrey B.
- Journal of Plasma Physics, Vol. 82, Issue 6

##
Generalized action invariants for drift waves-zonal flow systems

journal, December 1999

- Smolyakov, A. I.; Diamond, P. H.
- Physics of Plasmas, Vol. 6, Issue 12

##
Coherent nonlinear structures of drift wave turbulence modulated by zonal flows

journal, December 2001

- Kaw, Predhiman; Singh, Raghvendra; Diamond, P. H.
- Plasma Physics and Controlled Fusion, Vol. 44, Issue 1

##
On the stability of drift wave spectra with respect to zonal flow excitation

journal, May 2001

- Malkov, M. A.; Diamond, P. H.; Smolyakov, A.
- Physics of Plasmas, Vol. 8, Issue 5

##
Spatio-temporal evolution of the L → I → H transition

journal, September 2012

- Miki, K.; Diamond, P. H.; Gürcan, Ö. D.
- Physics of Plasmas, Vol. 19, Issue 9

##
Self-Regulating Shear Flow Turbulence: A Paradigm for the *L* to *H* Transition

journal, April 1994

- Diamond, P. H.; Liang, Y. -M.; Carreras, B. A.
- Physical Review Letters, Vol. 72, Issue 16

##
Zonal Flows and Transient Dynamics of the $L\mathrm{\text{\u2212}}H$ Transition

journal, May 2003

- Kim, Eun-jin; Diamond, P. H.
- Physical Review Letters, Vol. 90, Issue 18

##
Predator prey oscillations in a simple cascade model of drift wave turbulence

journal, November 2011

- Berionni, V.; Gürcan, Ö. D.
- Physics of Plasmas, Vol. 18, Issue 11

##
Direct identification of predator-prey dynamics in gyrokinetic simulations

journal, September 2015

- Kobayashi, Sumire; Gürcan, Özgür D.; Diamond, Patrick H.
- Physics of Plasmas, Vol. 22, Issue 9

##
Stationary Spectrum of Strong Turbulence in Magnetized Nonuniform Plasma

journal, July 1977

- Hasegawa, Akira; Mima, Kunioki
- Physical Review Letters, Vol. 39, Issue 4

##
Interactions of disparate scales in drift-wave turbulence

journal, December 2000

- Krommes, John A.; Kim, Chang-Bae
- Physical Review E, Vol. 62, Issue 6

##
Developments in the gyrofluid approach to Tokamak turbulence simulations

journal, August 1993

- Hammett, G. W.; Beer, M. A.; Dorland, W.
- Plasma Physics and Controlled Fusion, Vol. 35, Issue 8

##
Formation of solitary zonal structures via the modulational instability of drift waves

journal, May 2019

- Zhou, Yao; Zhu, Hongxuan; Dodin, I. Y.
- Plasma Physics and Controlled Fusion, Vol. 61, Issue 7

##
The quench rule, Dimits shift, and eigenmode localization by small-scale zonal flows

journal, January 2012

- Kobayashi, Sumire; Rogers, Barrett N.
- Physics of Plasmas, Vol. 19, Issue 1

##
A non-negative Wigner-type distribution

journal, January 1976

- Cartwright, N. D.
- Physica A: Statistical Mechanics and its Applications, Vol. 83, Issue 1

##
On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows

journal, August 2018

- Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
- Physics of Plasmas, Vol. 25, Issue 8

##
Bifurcation in electrostatic resistive drift wave turbulence

journal, October 2007

- Numata, Ryusuke; Ball, Rowena; Dewar, Robert L.
- Physics of Plasmas, Vol. 14, Issue 10

##
Dynamics of zonal flow saturation in strong collisionless drift wave turbulence

journal, November 2002

- Kim, Eun-jin; Diamond, P. H.
- Physics of Plasmas, Vol. 9, Issue 11

##
On the tertiary instability formalism of zonal flows in magnetized plasmas

journal, May 2018

- Rath, F.; Peeters, A. G.; Buchholz, R.
- Physics of Plasmas, Vol. 25, Issue 5

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*