Zonal Jet Creation from Secondary Instability of Drift Waves for Plasma Edge Turbulence

Abstract

A new strategy is presented to explain the creation and persistence of zonal flows widely observed in plasma edge turbulence. The core physics in the edge regime of the magnetic-fusion tokamaks can be described qualitatively by the one-state modified Hasegawa-Mima (MHM for short) model, which creates enhanced zonal flows and more physically relevant features in comparison with the familiar Charney-Hasegawa-Mima (CHM for short) model for both plasma and geophysical flows. The generation mechanism of zonal jets is displayed from the secondary instability analysis via nonlinear interactions with a background base state. Strong exponential growth in the zonal modes is induced due to a non-zonal drift wave base state in the MHM model, while stabilizing damping effect is shown with a zonal flow base state. Together with the selective decay effect from the dissipation, the secondary instability offers a complete characterization of the convergence process to the purely zonal structure. Direct numerical simulations with and without dissipation are carried out to confirm the instability theory. It shows clearly the emergence of a dominant zonal flow from pure non-zonal drift waves with small perturbation in the initial configuration. In comparison, the CHM model does not create instability in the zonal modes and usually converges to homogeneous turbulence.

Keywords

Zonal flow generation / Drift wave turbulence / Secondary instability / Modified Hasegawa-Mima model

References

[1] Dewar R L, Abdullatif R F. Zonal flow generation by modulational instability, Frontiers in Turbulence and Coherent Structures, 2007, Hackensack, NJ: World Scientific 415-430

[2] Diamond P H, Itoh S, Itoh K, Hahm T. Zonal flows in plasma—a review. Plasma Physics and Controlled Fusion, 2005, 47(5): R35-R161

[3] Fujisawa A. A review of zonal flow experiments. Nuclear Fusion, 2009, 49(1): 013001

[4] Hasegawa A, Mima K. Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. The Physics of Fluids, 1978, 21(1): 87-92

[5] Hasegawa A, Wakatani M. Plasma edge turbulence. Physical Review Letters, 1983, 50(9): 682

[6] Horton W. Drift waves and transport. Rev. Mod. Phys., 1999, 71: 735-778

[7] Lee Y, Smith L M. Stability of rossby waves in the fi-plane approximation. Physica D: Nonlinear Phenomena, 2003, 179(1–2): 53-91

[8] Lin Z, Hahm T S, Lee W W Turbulent transport reduction by zonal flows: Massively parallel simulations. Science, 1998, 281(5384): 1835-1837

[9] Majda A. Introduction to PDEs and Waves for the Atmosphere and Ocean, 9, 2003, Providence, RI: American Mathematical Soc

[10] Majda A J. Introduction to Turbulent Dynamical Systems in Complex Systems, 2016, Switzerland: Springer-Verlag

[11] Majda A J, Qi D. Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems. SIAM Review, 2018, 60(3): 491-549

[12] Majda A J, Qi D, Cerfon A J. A flux-balanced fluid model for collisional plasma edge turbulence: Model derivation and basic physical features. Physics of Plasmas, 2018, 25(10): 102307

[13] Majda A J, Shim S Y, Wang X. Selective decay for geophysical flows. Methods and applications of analysis, 2000, 7(3): 511-554

[14] Manfredi G, Roach C, Dendy R. Zonal flow and streamer generation in drift turbulence. Plasma Physics and Controlled Fusion, 2001, 43(6): 825

[15] Manz P, Ramisch M, Stroth U. Physical mechanism behind zonal-flow generation in drift-wave turbulence. Phys. Rev. Lett., 2009, 103: 165004

[16] Meshalkin L. Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech., 1962, 25: 1700

[17] Numata R, Ball R, Dewar R L. Bifurcation in electrostatic resistive drift wave turbulence. Physics of Plasmas, 2007, 14(10): 102312

[18] Pedlosky J. Geophysical Fluid Dynamics, 1987, New York: Springer-Verlag

[19] Pushkarev A V, Bos W J T, Nazarenko S V. Zonal flow generation and its feedback on turbulence production in drift wave turbulence. Physics of Plasmas, 2013, 20(4): 042304

[20] Qi D, Majda A J. Low-dimensional reduced-order models for statistical response and uncertainty quantification: Two-layer baroclinic turbulence. Journal of the Atmospheric Sciences, 2016, 73(12): 4609-4639

[21] Qi D, Majda A J. Transient metastability and selective decay for the coherent zonal structures in plasma drift wave turbulence. Journal of Nonlinear Science, 2019 1-43

[22] Qi D, Majda A J, Cerfon A J. A Flux-Balanced Fluid Model for Collisional Plasma Edge Turbulence: Numerical Simulations with Different Aspect Ratios. Physics of Plasmas, 2018

[23] Rhines P B. Waves and turbulence on a beta-plane. Journal of Fluid Mechanics, 1975, 69(3): 417-443

[24] Smolyakov A, Diamond P, Malkov M. Coherent structure phenomena in drift wave-zonal flow turbulence. Physical Review Letters, 2000, 84(3): 491

[25] Xanthopoulos P, Mischchenko A, Helander P Zonal flow dynamics and control of turbulent transport in stellarators. Phys. Rev. Lett., 2011, 107: 245002

[26] Zhu H, Zhou Y, Dodin I. On the structure of the drifton phase space and its relation to the rayleigh-kuo criterion of the zonal-flow stability. Physics of Plasmas, 2018, 25(7): 072121

[27] Zhu H, Zhou Y, Dodin I. On the rayleigh–kuo criterion for the tertiary instability of zonal flows. Physics of Plasmas, 2018, 25(8): 082121