skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Abstract

This study reports the stability conditions for intense zonal flows (ZFs) and the growth rate γ TI of the corresponding “tertiary” instability (TI) within the generalized Hasegawa–Mima plasma model. The analytical calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. 6, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytical formula for γ TI is derived and compared with numerical calculations. It is shown that, within the generalized Hasegawa–Mima model, a sinusoidal ZF is TI-unstable if and only if it satisfies the Rayleigh–Kuo criterion (known from geophysics) and that the ZF wave number exceeds the inverse ion sound radius. For non-sinusoidal ZFs, the results are qualitatively similar. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the perturbation wavelength is small compared to the ZF scale. Finally, this also means that the traditional wave kinetic equation, which is derived under the geometrical-optics assumption, cannot adequately describe the ZF stability.

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [1]
  1. Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences; Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
  2. Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
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:
1467162
Alternate Identifier(s):
OSTI ID: 1466870
Grant/Contract Number:  
[AC02-09CH11466]
Resource Type:
Accepted Manuscript
Journal Name:
Physics of Plasmas
Additional Journal Information:
[ Journal Volume: 25; Journal Issue: 8]; Journal ID: ISSN 1070-664X
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; perturbation theory; geometrical optics; plasma flows; geophysics; hydrodynamics

Citation Formats

Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. United States: N. p., 2018. Web. doi:10.1063/1.5038859.
Zhu, Hongxuan, Zhou, Yao, & Dodin, I. Y. On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. United States. doi:10.1063/1.5038859.
Zhu, Hongxuan, Zhou, Yao, and Dodin, I. Y. Mon . "On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows". United States. doi:10.1063/1.5038859. https://www.osti.gov/servlets/purl/1467162.
@article{osti_1467162,
title = {On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows},
author = {Zhu, Hongxuan and Zhou, Yao and Dodin, I. Y.},
abstractNote = {This study reports the stability conditions for intense zonal flows (ZFs) and the growth rate γTI of the corresponding “tertiary” instability (TI) within the generalized Hasegawa–Mima plasma model. The analytical calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. 6, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytical formula for γTI is derived and compared with numerical calculations. It is shown that, within the generalized Hasegawa–Mima model, a sinusoidal ZF is TI-unstable if and only if it satisfies the Rayleigh–Kuo criterion (known from geophysics) and that the ZF wave number exceeds the inverse ion sound radius. For non-sinusoidal ZFs, the results are qualitatively similar. As a corollary, there is no TI in the geometrical-optics limit, i.e., when the perturbation wavelength is small compared to the ZF scale. Finally, this also means that the traditional wave kinetic equation, which is derived under the geometrical-optics assumption, cannot adequately describe the ZF stability.},
doi = {10.1063/1.5038859},
journal = {Physics of Plasmas},
number = [8],
volume = [25],
place = {United States},
year = {2018},
month = {8}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 5 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Noncurvature-driven modes in a transport barrier
journal, June 2005

  • Rogers, B. N.; Dorland, W.
  • Physics of Plasmas, Vol. 12, Issue 6
  • DOI: 10.1063/1.1928250

Stellarator Turbulence: Subdominant Eigenmodes and Quasilinear Modeling
journal, February 2016


Another look at zonal flows: Resonance, shearing, and frictionless saturation
journal, April 2018

  • Li, J. C.; Diamond, P. H.
  • Physics of Plasmas, Vol. 25, Issue 4
  • DOI: 10.1063/1.5027107

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
  • DOI: 10.1063/1.4753931

Zonal flow as pattern formation
journal, October 2013

  • Parker, Jeffrey B.; Krommes, John A.
  • Physics of Plasmas, Vol. 20, Issue 10
  • DOI: 10.1063/1.4828717

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
  • DOI: 10.1063/1.4971813

Interactions of disparate scales in drift-wave turbulence
journal, December 2000


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
  • DOI: 10.1063/1.5026670

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
  • DOI: 10.1175/JAS4016.1

Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations
journal, November 2016


Electron temperature gradient driven turbulence
journal, May 2000

  • Jenko, F.; Dorland, W.; Kotschenreuther, M.
  • Physics of Plasmas, Vol. 7, Issue 5
  • DOI: 10.1063/1.874014

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
  • DOI: 10.1063/1.1514641

Coherent Structure Phenomena in Drift Wave–Zonal Flow Turbulence
journal, January 2000


The instability to long waves of unbounded parallel inviscid flow
journal, October 1962


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
  • DOI: 10.1063/1.1330204

On Instability of Parallel Flow of Inviscid Fluid in a Rotating System with Variable Coriolis Parameter Journal of Mathematics and Physics
journal, April 1964

  • Howard, Louis N.; Drazin, Philip G.
  • Journal of Mathematics and Physics, Vol. 43, Issue 1-4
  • DOI: 10.1002/sapm196443183

Generation of zonal flows through symmetry breaking of statistical homogeneity
journal, March 2014


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
  • DOI: 10.1063/1.873725

A review of zonal flow experiments
journal, December 2008


Electron Temperature Gradient Turbulence
journal, December 2000


Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation
journal, May 2018


Generation and Stability of Zonal Flows in Ion-Temperature-Gradient Mode Turbulence
journal, December 2000


Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra
journal, July 2016


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
  • DOI: 10.1063/1.3677355

Bifurcation in electrostatic resistive drift wave turbulence
journal, October 2007

  • Numata, Ryusuke; Ball, Rowena; Dewar, Robert L.
  • Physics of Plasmas, Vol. 14, Issue 10
  • DOI: 10.1063/1.2796106

Turbulent Transport Reduction by Zonal Flows: Massively Parallel Simulations
journal, September 1998


On non-local energy transfer via zonal flow in the Dimits shift
journal, October 2017


An analytic model for limiting high density LH transition by the onset of the tertiary instability
journal, July 2016

  • Singh, Raghvendra; Jhang, Hogun; Kaang, Helen H.
  • Physics of Plasmas, Vol. 23, Issue 7
  • DOI: 10.1063/1.4958646

Zonal flows in plasma—a review
journal, April 2005


Influence of sheared poloidal rotation on edge turbulence
journal, January 1990

  • Biglari, H.; Diamond, P. H.; Terry, P. W.
  • Physics of Fluids B: Plasma Physics, Vol. 2, Issue 1
  • DOI: 10.1063/1.859529

A stochastic structural stability theory model of the drift wave–zonal flow system
journal, November 2009

  • Farrell, Brian F.; Ioannou, Petros J.
  • Physics of Plasmas, Vol. 16, Issue 11
  • DOI: 10.1063/1.3258666

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
  • DOI: 10.1063/1.873950

Coupling of damped and growing modes in unstable shear flow
journal, June 2017

  • Fraser, A. E.; Terry, P. W.; Zweibel, E. G.
  • Physics of Plasmas, Vol. 24, Issue 6
  • DOI: 10.1063/1.4985322

Zonostrophic Instability
journal, May 2012

  • Srinivasan, Kaushik; Young, W. R.
  • Journal of the Atmospheric Sciences, Vol. 69, Issue 5
  • DOI: 10.1175/JAS-D-11-0200.1

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
  • DOI: 10.1063/1.5039652

Stationary Spectrum of Strong Turbulence in Magnetized Nonuniform Plasma
journal, July 1977


Developments in the gyrofluid approach to Tokamak turbulence simulations
journal, August 1993


Self-Regulating Shear Flow Turbulence: A Paradigm for the L to H Transition
journal, April 1994


Zonal Flows and Transient Dynamics of the L H Transition
journal, May 2003


The barotropic stability of the mean winds in the atmosphere
journal, March 1962


Statistical State Dynamics of Jet–Wave Coexistence in Barotropic Beta-Plane Turbulence
journal, May 2016

  • Constantinou, Navid C.; Farrell, Brian F.; Ioannou, Petros J.
  • Journal of the Atmospheric Sciences, Vol. 73, Issue 5
  • DOI: 10.1175/JAS-D-15-0288.1

On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]
dataset, May 2018

  • Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
  • Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 Files (12.2 kB)
  • DOI: 10.5281/zenodo.1241545

On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]
dataset, May 2018

  • Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
  • Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 files (12.2 kB)
  • DOI: 10.5281/zenodo.1241546

    Works referencing / citing this record:

    On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]
    dataset, May 2018

    • Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
    • Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 Files (12.2 kB)
    • DOI: 10.5281/zenodo.1241545

    On The Rayleigh--Kuo Criterion For The Tertiary Instability Of Zonal Flows [Supplementary Data]
    dataset, May 2018

    • Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y.
    • Zenodo-Supplementary information for journal article at DOI: 10.1063/1.5038859, 5 files (12.2 kB)
    • DOI: 10.5281/zenodo.1241546

    Reduction of edge plasma turbulence via cross-phase decrease by zonal fields
    journal, June 2019

    • Kim, Chang-Bae; An, Chan-Yong; Min, Byunghoon
    • Plasma Physics and Controlled Fusion, Vol. 61, Issue 8
    • DOI: 10.1088/1361-6587/ab2973