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

Ruiz, Daniel Edward; Glinsky, Michael Edwin; Dodin, Ilya  Y.

doi:10.1017/S0022377818001307

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

Journal Article · Mon Jan 07 23:00:00 EST 2019 · Journal of Plasma Physics

DOI:https://doi.org/10.1017/S0022377818001307· OSTI ID:1492364

^[1]; Glinsky, Michael Edwin ^[1]; Dodin, Ilya Y. ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Princeton Univ., Princeton, NJ (United States)

Here, the formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is often described using statistical theories derived within the quasilinear approximation. However, this approximation neglects wave–wave collisions. Hence, some important effects such as the Batchelor–Kraichnan inverse-energy cascade are not captured within this approach. Here we derive a wave kinetic equation that includes a DW collision operator in the presence of zonal flows. Our derivation makes use of the Weyl calculus, the quasinormal statistical closure and the geometrical-optics approximation. The obtained model conserves both the total enstrophy and energy of the system. The derived DW collision operator breaks down at the Rayleigh–Kuo threshold. This threshold is missed by homogeneous-turbulence theory but expected from a full-wave quasilinear analysis. In the future, this theory might help better understand the interactions between drift waves and zonal flows, including the validity domain of the quasilinear approximation that is commonly used in the literature.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1492364

Alternate ID(s):: OSTI ID: 1511444

Report Number(s):: SAND--2018-13584J; 670665

Journal Information:: Journal of Plasma Physics, Journal Name: Journal of Plasma Physics Journal Issue: 01 Vol. 85; ISSN applab; ISSN 0022-3778

Publisher:: Cambridge University PressCopyright Statement

Country of Publication:: United States

Language:: English

References (59)

Wave Turbulence Nazarenko, Sergey Lecture Notes in Physics (book series) https://doi.org/10.1007/978-3-642-15942-8	book	January 2011
Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves McDonald, Steven W. Physics Reports, Vol. 158, Issue 6 https://doi.org/10.1016/0370-1573(88)90012-9	journal	February 1988
Fundamental statistical descriptions of plasma turbulence in magnetic fields Krommes, John A. Physics Reports, Vol. 360, Issue 1-4 https://doi.org/10.1016/S0370-1573(01)00066-7	journal	April 2002
Geometric view on noneikonal waves Dodin, I. Y. Physics Letters A, Vol. 378, Issue 22-23 https://doi.org/10.1016/j.physleta.2014.04.004	journal	April 2014
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
Ray Tracing and Beyond Tracy, E. R.; Brizard, A. J.; Richardson, A. S. Cambridge University Press https://doi.org/10.1017/CBO9780511667565	book	January 2014
Turbulence: The Legacy of A. N. Kolmogorov Frisch, Uriel https://doi.org/10.1017/CBO9781139170666	book	January 1996
A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence Ogura, Yoshimitsu Journal of Fluid Mechanics, Vol. 16, Issue 01 https://doi.org/10.1017/S0022112063000562	journal	May 1963
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
Quantum mechanics as a statistical theory Moyal, J. E. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 45, Issue 1 https://doi.org/10.1017/S0305004100000487	journal	January 1949
Three-dimensional rotating Couette flow via the generalised quasilinear approximation Tobias, S. M.; Marston, J. B. Journal of Fluid Mechanics, Vol. 810 https://doi.org/10.1017/jfm.2016.727	journal	November 2016
Bifurcation and scaling of drift wave turbulence intensity with collisional zonal flow damping Malkov, M. A.; Diamond, P. H. Physics of Plasmas, Vol. 8, Issue 9 https://doi.org/10.1063/1.1394760	journal	September 2001
On the nature of bursting in transport and turbulence in drift wave–zonal flow systems Malkov, M. A.; Diamond, P. H.; Rosenbluth, M. N. Physics of Plasmas, Vol. 8, Issue 12 https://doi.org/10.1063/1.1415424	journal	December 2001
Wigner Method in Quantum Statistical Mechanics İmre, Kaya; Özizmir, Ercüment; Rosenbaum, Marcos Journal of Mathematical Physics, Vol. 8, Issue 5 https://doi.org/10.1063/1.1705323	journal	May 1967
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
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
Coherent structures in ion temperature gradient turbulence-zonal flow Singh, Rameswar; Singh, R.; Kaw, P. Physics of Plasmas, Vol. 21, Issue 10 https://doi.org/10.1063/1.4898207	journal	October 2014
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
Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics Parker, Jeffrey B. Physics of Plasmas, Vol. 25, Issue 5 https://doi.org/10.1063/1.5018142	journal	May 2018
Influence of sheared poloidal rotation on edge turbulence Biglari, H.; Diamond, P. H.; Terry, P. W. Physics of Fluids B: Plasma Physics, Vol. 2, Issue 1 https://doi.org/10.1063/1.859529	journal	January 1990
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
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
Wave turbulence Nazarenko, Sergey Contemporary Physics, Vol. 56, Issue 3 https://doi.org/10.1080/00107514.2015.1015250	journal	March 2015
A review of zonal flow experiments Fujisawa, Akihide Nuclear Fusion, Vol. 49, Issue 1 https://doi.org/10.1088/0029-5515/49/1/013001	journal	December 2008
Review of developments in turbulence theory Leslie, D. C. Reports on Progress in Physics, Vol. 36, Issue 11 https://doi.org/10.1088/0034-4885/36/11/001	journal	November 1973
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
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
Direct measurement of zonal flows and geodesic acoustic mode oscillations in ASDEX Upgrade using Doppler reflectometry Conway, G. D.; Scott, B.; Schirmer, J. Plasma Physics and Controlled Fusion, Vol. 47, Issue 8 https://doi.org/10.1088/0741-3335/47/8/003	journal	July 2005
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
Cumulant expansions for atmospheric flows Ait-Chaalal, Farid; Schneider, Tapio; Meyer, Bettina New Journal of Physics, Vol. 18, Issue 2 https://doi.org/10.1088/1367-2630/18/2/025019	journal	February 2016
Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space Baker, George A. Physical Review, Vol. 109, Issue 6 https://doi.org/10.1103/PhysRev.109.2198	journal	March 1958
On the Quantum Correction For Thermodynamic Equilibrium Wigner, E. Physical Review, Vol. 40, Issue 5 https://doi.org/10.1103/PhysRev.40.749	journal	June 1932
Weyl representation for electromagnetic waves: The wave kinetic equation McDonald, Steven W.; Kaufman, Allan N. Physical Review A, Vol. 32, Issue 3 https://doi.org/10.1103/PhysRevA.32.1708	journal	September 1985
Wave kinetic equation in a fluctuating medium McDonald, Steven W. Physical Review A, Vol. 43, Issue 8 https://doi.org/10.1103/PhysRevA.43.4484	journal	April 1991
Loss spectra of graphite-related systems: A multiwall carbon nanotube, a single-wall carbon nanotube bundle, and graphite layers Shyu, F. L.; Lin, M. F. Physical Review B, Vol. 62, Issue 12 https://doi.org/10.1103/PhysRevB.62.8508	journal	September 2000
Direct Statistical Simulation of Out-of-Equilibrium Jets Tobias, S. M.; Marston, J. B. Physical Review Letters, Vol. 110, Issue 10 https://doi.org/10.1103/PhysRevLett.110.104502	journal	March 2013
Stationary Zonal Flows during the Formation of the Edge Transport Barrier in the JET Tokamak Hillesheim, J. C.; Delabie, E.; Meyer, H. Physical Review Letters, Vol. 116, Issue 6 https://doi.org/10.1103/PhysRevLett.116.065002	journal	February 2016
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
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
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
Electron Temperature Gradient Turbulence Dorland, W.; Jenko, F.; Kotschenreuther, M. Physical Review Letters, Vol. 85, Issue 26 https://doi.org/10.1103/PhysRevLett.85.5579	journal	December 2000
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
Quasiparticle Approach to the Modulational Instability of Drift Waves Coupling to Zonal Flows Trines, R.; Bingham, R.; Silva, L. O. Physical Review Letters, Vol. 94, Issue 16 https://doi.org/10.1103/PhysRevLett.94.165002	journal	April 2005
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
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
Structural Stability of Turbulent Jets Farrell, Brian F.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 60, Issue 17 https://doi.org/10.1175/1520-0469(2003)060<2101:SSOTJ>2.0.CO;2	journal	September 2003
Statistics of an Unstable Barotropic Jet from a Cumulant Expansion Marston, J. B.; Conover, E.; Schneider, Tapio Journal of the Atmospheric Sciences, Vol. 65, Issue 6 https://doi.org/10.1175/2007JAS2510.1	journal	June 2008
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
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
Corrections to classical radiometry Littlejohn, Robert G.; Winston, Roland Journal of the Optical Society of America A, Vol. 10, Issue 9 https://doi.org/10.1364/JOSAA.10.002024	journal	January 1993
Cumulant expansions for atmospheric flows Ait-Chaalal, Farid; Schneider, Tapio; Meyer, Bettina ETH Zurich https://doi.org/10.3929/ethz-b-000114310	text	January 2016
Statistics of an Unstable Barotropic Jet from a Cumulant Expansion Marston, J. B.; Conover, E.; Schneider, Tapio arXiv https://doi.org/10.48550/arxiv.0705.0011	text	January 2007
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
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
Cumulant expansions for atmospheric flows Ait-Chaalal, Farid; Schneider, Tapio; Meyer, Bettina arXiv https://doi.org/10.48550/arxiv.1505.07643	text	January 2015
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
Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics Parker, Jeffrey B. arXiv https://doi.org/10.48550/arxiv.1803.08643	text	January 2018
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

Cited By (4)

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 saturation and oscillations of collisionless zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. New Journal of Physics, Vol. 21, Issue 6 https://doi.org/10.1088/1367-2630/ab2251	journal	June 2019
Nonlinear saturation and oscillations of collisionless zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. Zenodo https://doi.org/10.5281/zenodo.2563448	text	January 2019
Nonlinear saturation and oscillations of collisionless zonal flows Zhu, Hongxuan; Zhou, Yao; Dodin, I. Y. Zenodo https://doi.org/10.5281/zenodo.2563449	text	January 2019

Similar Records

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

Journal Article · Tue May 29 00:00:00 EDT 2018 · Physical Review E · OSTI ID:1440782

Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics

Journal Article · Thu Mar 22 00:00:00 EDT 2018 · Physics of Plasmas · OSTI ID:1466924

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY

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

Citation Formats

References (59)

Cited By (4)

Similar Records

Related Subjects