Nonlinear saturation and oscillations of collisionless zonal flows

Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.

doi:10.1088/1367-2630/ab2251

Title: Nonlinear saturation and oscillations of collisionless zonal flows

Journal Article · 06 June 2019 · New Journal of Physics

DOI: https://doi.org/10.1088/1367-2630/ab2251 · OSTI ID:1525338

^[1];

^[2];

^[1]

Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Princeton Univ., NJ (United States)
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

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.

View Journal Article

Cite

Export

Save

Research Organization:: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: AC02-09CH11466

OSTI ID:: 1525338

Alternate ID(s):: OSTI ID: 1544248

Journal Information:: New Journal of Physics, Vol. 21, Issue 6; ISSN 1367-2630

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 7 works

Citation information provided by
Web of Science

References (71)

Zonal flow and streamer generation in drift turbulence Manfredi, G.; Roach, C. M.; Dendy, R. O. Plasma Physics and Controlled Fusion, Vol. 43, Issue 6 https://doi.org/10.1088/0741-3335/43/6/307	journal	May 2001
The stability of planetary waves on an infinite beta‐plane Gill, A. E. Geophysical Fluid Dynamics, Vol. 6, Issue 1 https://doi.org/10.1080/03091927409365786	journal	January 1974
Spatio-temporal evolution of the L → I → H transition Miki, K.; Diamond, P. H.; Gürcan, Ö. D. Physics of Plasmas, Vol. 19, Issue 9 https://doi.org/10.1063/1.4753931	journal	September 2012
Zonal-flow dynamics from a phase-space perspective Ruiz, D. E.; Parker, J. B.; Shi, E. L. Physics of Plasmas, Vol. 23, Issue 12 https://doi.org/10.1063/1.4971813	journal	December 2016
Interactions of disparate scales in drift-wave turbulence Krommes, John A.; Kim, Chang-Bae Physical Review E, Vol. 62, Issue 6 https://doi.org/10.1103/PhysRevE.62.8508	journal	December 2000
On the tertiary instability formalism of zonal flows in magnetized plasmas Rath, F.; Peeters, A. G.; Buchholz, R. Physics of Plasmas, Vol. 25, Issue 5 https://doi.org/10.1063/1.5026670	journal	May 2018
A non-negative Wigner-type distribution Cartwright, N. D. Physica A: Statistical Mechanics and its Applications, Vol. 83, Issue 1 https://doi.org/10.1016/0378-4371(76)90145-X	journal	January 1976
Structure and Spacing of Jets in Barotropic Turbulence Farrell, Brian F.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 64, Issue 10 https://doi.org/10.1175/JAS4016.1	journal	October 2007
On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. Physics of Plasmas, Vol. 25, Issue 8 https://doi.org/10.1063/1.5038859	journal	August 2018
Electron temperature gradient driven turbulence Jenko, F.; Dorland, W.; Kotschenreuther, M. Physics of Plasmas, Vol. 7, Issue 5 https://doi.org/10.1063/1.874014	journal	May 2000
Dynamics of zonal flow saturation in strong collisionless drift wave turbulence Kim, Eun-jin; Diamond, P. H. Physics of Plasmas, Vol. 9, Issue 11 https://doi.org/10.1063/1.1514641	journal	November 2002
Generation of zonal flows through symmetry breaking of statistical homogeneity Parker, Jeffrey B.; Krommes, John A. New Journal of Physics, Vol. 16, Issue 3 https://doi.org/10.1088/1367-2630/16/3/035006	journal	March 2014
Generalized action invariants for drift waves-zonal flow systems Smolyakov, A. I.; Diamond, P. H. Physics of Plasmas, Vol. 6, Issue 12 https://doi.org/10.1063/1.873725	journal	December 1999
S3T Stability of the Homogeneous State of Barotropic Beta-Plane Turbulence Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 72, Issue 5 https://doi.org/10.1175/JAS-D-14-0213.1	journal	May 2015
The modulational instability in the extended Hasegawa-Mima equation with a finite Larmor radius Gallagher, S.; Hnat, B.; Connaughton, C. Physics of Plasmas, Vol. 19, Issue 12 https://doi.org/10.1063/1.4773050	journal	December 2012
Streamer and zonal flow generation from envelope modulations in drift wave turbulence Champeaux, S.; Diamond, P. H. Physics Letters A, Vol. 288, Issue 3-4 https://doi.org/10.1016/S0375-9601(01)00549-7	journal	September 2001
The oscillation between Rossby wave and zonal flow in a barotropic fluid Mahanti, A. C. Archives for Meteorology, Geophysics, and Bioclimatology Series A, Vol. 30, Issue 3 https://doi.org/10.1007/BF02257844	journal	September 1981
Generation and Stability of Zonal Flows in Ion-Temperature-Gradient Mode Turbulence Rogers, B. N.; Dorland, W.; Kotschenreuther, M. Physical Review Letters, Vol. 85, Issue 25 https://doi.org/10.1103/PhysRevLett.85.5336	journal	December 2000
The quench rule, Dimits shift, and eigenmode localization by small-scale zonal flows Kobayashi, Sumire; Rogers, Barrett N. Physics of Plasmas, Vol. 19, Issue 1 https://doi.org/10.1063/1.3677355	journal	January 2012
Nonlinear interactions between drift waves and zonal flows Shukla, P. K.; Stenflo, L. The European Physical Journal D - Atomic, Molecular and Optical Physics, Vol. 20, Issue 1 https://doi.org/10.1140/epjd/e2002-00119-x	journal	July 2002
Coherent nonlinear structures of drift wave turbulence modulated by zonal flows Kaw, Predhiman; Singh, Raghvendra; Diamond, P. H. Plasma Physics and Controlled Fusion, Vol. 44, Issue 1 https://doi.org/10.1088/0741-3335/44/1/305	journal	December 2001
Streamer saturation: a dynamical systems approach Jenko, F. Physics Letters A, Vol. 351, Issue 6 https://doi.org/10.1016/j.physleta.2005.11.029	journal	March 2006
On non-local energy transfer via zonal flow in the Dimits shift St-Onge, Denis A. Journal of Plasma Physics, Vol. 83, Issue 5 https://doi.org/10.1017/S0022377817000708	journal	October 2017
Predator prey oscillations in a simple cascade model of drift wave turbulence Berionni, V.; Gürcan, Ö. D. Physics of Plasmas, Vol. 18, Issue 11 https://doi.org/10.1063/1.3656953	journal	November 2011
Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation Ruiz, D. E.; Glinsky, M. E.; Dodin, I. Y. Journal of Plasma Physics, Vol. 85, Issue 1 https://doi.org/10.1017/S0022377818001307	journal	January 2019
Excitation of zonal flow by drift waves in toroidal plasmas Chen, Liu; Lin, Zhihong; White, Roscoe Physics of Plasmas, Vol. 7, Issue 8 https://doi.org/10.1063/1.874222	journal	August 2000
Zonal flows in plasma—a review Diamond, P. H.; Itoh, S-I; Itoh, K. Plasma Physics and Controlled Fusion, Vol. 47, Issue 5 https://doi.org/10.1088/0741-3335/47/5/R01	journal	April 2005
On the structure of the drifton phase space and its relation to the Rayleigh–Kuo criterion of the zonal-flow stability Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. Physics of Plasmas, Vol. 25, Issue 7 https://doi.org/10.1063/1.5039652	journal	July 2018
Developments in the gyrofluid approach to Tokamak turbulence simulations Hammett, G. W.; Beer, M. A.; Dorland, W. Plasma Physics and Controlled Fusion, Vol. 35, Issue 8 https://doi.org/10.1088/0741-3335/35/8/006	journal	August 1993
Rossby and drift wave turbulence and zonal flows: The Charney–Hasegawa–Mima model and its extensions Connaughton, Colm; Nazarenko, Sergey; Quinn, Brenda Physics Reports, Vol. 604 https://doi.org/10.1016/j.physrep.2015.10.009	journal	December 2015
Self-Regulating Shear Flow Turbulence: A Paradigm for the L to H Transition Diamond, P. H.; Liang, Y. -M.; Carreras, B. A. Physical Review Letters, Vol. 72, Issue 16 https://doi.org/10.1103/PhysRevLett.72.2565	journal	April 1994
Influence of the mean flow on zonal flow generation Lashkin, Volodymyr M. Physics of Plasmas, Vol. 15, Issue 12 https://doi.org/10.1063/1.3050064	journal	December 2008
Zonostrophic instability driven by discrete particle noise St-Onge, D. A.; Krommes, J. A. Physics of Plasmas, Vol. 24, Issue 4 https://doi.org/10.1063/1.4978786	journal	April 2017
Nonlinear damping of zonal flows Koshkarov, O.; Smolyakov, A. I.; Mendonca, J. T. Plasma Physics Reports, Vol. 42, Issue 8 https://doi.org/10.1134/S1063780X16080067	journal	August 2016
Zonal flow as pattern formation Parker, Jeffrey B.; Krommes, John A. Physics of Plasmas, Vol. 20, Issue 10 https://doi.org/10.1063/1.4828717	journal	October 2013
Magnetic Suppression of Zonal Flows on a Beta Plane Constantinou, Navid C.; Parker, Jeffrey B. The Astrophysical Journal, Vol. 863, Issue 1 https://doi.org/10.3847/1538-4357/aace53	journal	August 2018
Direct identification of predator-prey dynamics in gyrokinetic simulations Kobayashi, Sumire; Gürcan, Özgür D.; Diamond, Patrick H. Physics of Plasmas, Vol. 22, Issue 9 https://doi.org/10.1063/1.4930127	journal	September 2015
Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations Parker, Jeffrey B. Journal of Plasma Physics, Vol. 82, Issue 6 https://doi.org/10.1017/S0022377816001021	journal	November 2016
Zonal flow and streamer generation in drift turbulence Manfredi, G.; Roach, C. M.; Dendy, R. O. Plasma Physics and Controlled Fusion, Vol. 43, Issue 7 https://doi.org/10.1088/0741-3335/43/7/402	journal	June 2001
On the relation between secondary and modulational instabilities Strintzi, D.; Jenko, F. Physics of Plasmas, Vol. 14, Issue 4 https://doi.org/10.1063/1.2720370	journal	April 2007
Coherent Structure Phenomena in Drift Wave–Zonal Flow Turbulence Smolyakov, A. I.; Diamond, P. H.; Malkov, M. Physical Review Letters, Vol. 84, Issue 3 https://doi.org/10.1103/PhysRevLett.84.491	journal	January 2000
On the stability of drift wave spectra with respect to zonal flow excitation Malkov, M. A.; Diamond, P. H.; Smolyakov, A. Physics of Plasmas, Vol. 8, Issue 5 https://doi.org/10.1063/1.1330204	journal	May 2001
Jovian atmospheric dynamics: an update after Galileo and Cassini Vasavada, Ashwin R.; Showman, Adam P. Reports on Progress in Physics, Vol. 68, Issue 8 https://doi.org/10.1088/0034-4885/68/8/R06	journal	July 2005
Nonlinear saturation and oscillations of collisionless zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States) https://doi.org/10.11578/1562105	dataset	January 2019
The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows Lashmore-Davies, C. N.; McCarthy, D. R.; Thyagaraja, A. Physics of Plasmas, Vol. 8, Issue 12 https://doi.org/10.1063/1.1416881	journal	December 2001
Modulational instability of Rossby and drift waves and generation of zonal jets Connaughton, Colm P.; Nadiga, Balasubramanya T.; Nazarenko, Sergey V. Journal of Fluid Mechanics, Vol. 654 https://doi.org/10.1017/S0022112010000510	journal	May 2010
Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation Zhu, Hongxuan; Zhou, Yao; Ruiz, D. E. Physical Review E, Vol. 97, Issue 5 https://doi.org/10.1103/PhysRevE.97.053210	journal	May 2018
Bifurcation in electrostatic resistive drift wave turbulence Numata, Ryusuke; Ball, Rowena; Dewar, Robert L. Physics of Plasmas, Vol. 14, Issue 10 https://doi.org/10.1063/1.2796106	journal	October 2007
Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations Lin, Z. Science, Vol. 281, Issue 5384 https://doi.org/10.1126/science.281.5384.1835	journal	September 1998
Zonal flow generation by parametric instability in magnetized plasmas and geostrophic fluids Smolyakov, A. I.; Diamond, P. H.; Shevchenko, V. I. Physics of Plasmas, Vol. 7, Issue 5 https://doi.org/10.1063/1.873950	journal	May 2000
Generalized Quasilinear Approximation: Application to Zonal Jets Marston, J. B.; Chini, G. P.; Tobias, S. M. Physical Review Letters, Vol. 116, Issue 21 https://doi.org/10.1103/PhysRevLett.116.214501	journal	May 2016
Formation of solitary zonal structures via the modulational instability of drift waves Zhou, Yao; Zhu, Hongxuan; Dodin, I. Y. Plasma Physics and Controlled Fusion, Vol. 61, Issue 7 https://doi.org/10.1088/1361-6587/ab16a8	journal	May 2019
Nonlinear Damping of Zonal Flows Koshkarov, O.; Smolyakov, A. I.; Mendonca, J. T. Физика плазмы, Vol. 42, Issue 8 https://doi.org/10.7868/s0367292116080060	journal	January 2016
Zonostrophic Instability Srinivasan, Kaushik; Young, W. R. Journal of the Atmospheric Sciences, Vol. 69, Issue 5 https://doi.org/10.1175/JAS-D-11-0200.1	journal	May 2012
Stationary Spectrum of Strong Turbulence in Magnetized Nonuniform Plasma Hasegawa, Akira; Mima, Kunioki Physical Review Letters, Vol. 39, Issue 4 https://doi.org/10.1103/PhysRevLett.39.205	journal	July 1977
Zonal Flows and Transient Dynamics of the L − H Transition Kim, Eun-jin; Diamond, P. H. Physical Review Letters, Vol. 90, Issue 18 https://doi.org/10.1103/PhysRevLett.90.185006	journal	May 2003
On the effects of nonuniform zonal flow in the resistive-drift plasma Kim, Chang-Bae; Min, Byunghoon; An, Chan-Yong Plasma Physics and Controlled Fusion, Vol. 61, Issue 3 https://doi.org/10.1088/1361-6587/aaf622	journal	January 2019
Dynamic Instability of Two-Dimensional Nondivergent flow in a Barotropic Atmosphere Kuo, Hsiao-lan Journal of Meteorology, Vol. 6, Issue 2 https://doi.org/10.1175/1520-0469(1949)006<0105:DIOTDN>2.0.CO;2	journal	April 1949
Bifurcation in electrostatic resistive drift wave turbulence Numata, Ryusuke; Ball, Rowena; Dewar, Robert L. arXiv https://doi.org/10.48550/arxiv.0708.4317	text	January 2007
Influence of the mean flow on zonal flow generation Lashkin, V. M. arXiv https://doi.org/10.48550/arxiv.0808.3456	text	January 2008
Modulational instability of Rossby and drift waves and generation of zonal jets Connaughton, Colm; Nadiga, Balu; Nazarenko, Sergey arXiv https://doi.org/10.48550/arxiv.0905.2243	text	January 2009
Zonal Flow as Pattern Formation Parker, Jeffrey B.; Krommes, John A. arXiv https://doi.org/10.48550/arxiv.1301.5059	text	January 2013
Rossby and Drift Wave Turbulence and Zonal Flows: the Charney-Hasegawa-Mima model and its extensions Connaughton, Colm; Nazarenko, Sergey; Quinn, Brenda arXiv https://doi.org/10.48550/arxiv.1407.1896	text	January 2014
S3T stability of the homogeneous state of barotropic beta-plane turbulence Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J. arXiv https://doi.org/10.48550/arxiv.1407.3354	text	January 2014
Dynamics of Zonal Flows: Failure of Wave-Kinetic Theory, and New Geometrical Optics Approximations Parker, Jeffrey B. arXiv https://doi.org/10.48550/arxiv.1604.06904	text	January 2016
On Nonlocal Energy Transfer via Zonal Flow in the Dimits Shift St-Onge, Denis A. arXiv https://doi.org/10.48550/arxiv.1704.05406	text	January 2017
On the Rayleigh--Kuo criterion for the tertiary instability of zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. arXiv https://doi.org/10.48550/arxiv.1805.02233	text	January 2018
Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation Ruiz, D. E.; Glinsky, M. E.; Dodin, I. Y. arXiv https://doi.org/10.48550/arxiv.1901.02518	text	January 2019
Generalized Action Invariants for Drift Waves-Zonal Flow Systems Smolyakov, A. I.; Diamond, P. H. arXiv https://doi.org/10.48550/arxiv.physics/9910018	text	January 1999
The property of universality for wigglers and invariance of electromagnetic radiation Smolyakov, M. N. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 467-468 https://doi.org/10.1016/s0168-9002(01)00275-3	journal	July 2001
The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows Lashmore-Davies, C. N.; McCarthy, D. R.; Thyagaraja, A. Physics of Plasmas, Vol. 8, Issue 12 https://doi.org/10.1063/1.1416881	journal	December 2001