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Theory of the tertiary instability and the Dimits shift within a scalar model

Part of: Focus on Fusion

Published online by Cambridge University Press:  20 August 2020

Hongxuan Zhu Open the ORCID record for Hongxuan Zhu [Opens in a new window] ,
Yao Zhou Open the ORCID record for Yao Zhou [Opens in a new window]  and
I. Y. Dodin Open the ORCID record for I. Y. Dodin [Opens in a new window]
Hongxuan Zhu*
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ08544, USA
Yao Zhou
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ08543, USA
I. Y. Dodin
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

The Dimits shift is the shift between the threshold of the drift-wave primary instability and the actual onset of turbulent transport in a magnetized plasma. It is generally attributed to the suppression of turbulence by zonal flows, but developing a more detailed understanding calls for consideration of specific reduced models. The modified Terry–Horton system has been proposed by St-Onge (J. Plasma Phys., vol. 83, 2017, 905830504) as a minimal model capturing the Dimits shift. Here, we use this model to develop an analytic theory of the Dimits shift and a related theory of the tertiary instability of zonal flows. We show that tertiary modes are localized near extrema of the zonal velocity $U(x)$, where $x$ is the radial coordinate. By approximating $U(x)$ with a parabola, we derive the tertiary-instability growth rate using two different methods and show that the tertiary instability is essentially the primary drift-wave instability modified by the local $U'' \doteq {\rm d}^2 U/{\rm d} x^2 $. Then, depending on $U''$, the tertiary instability can be suppressed or unleashed. The former corresponds to the case when zonal flows are strong enough to suppress turbulence (Dimits regime), while the latter corresponds to the case when zonal flows are unstable and turbulence develops. This understanding is different from the traditional paradigm that turbulence is controlled by the flow shear $| {\rm d} U / {\rm d} x |$. Our analytic predictions are in agreement with direct numerical simulations of the modified Terry–Horton system.

Keywords

fusion plasmaplasma instabilitiesplasma waves
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 4 , August 2020 , 905860405
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Biglari, H., Diamond, P. & Terry, P. 1990 Influence of sheared poloidal rotation on edge turbulence. Phys. Fluids B: Plasma Phys. 2 (1), 1–4.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Courier Corporation.Google Scholar
Dewar, R. L. & Abdullatif, R. F. 2007 Zonal flow generation by modulational instability. In Frontiers in Turbulence and Coherent Structures (ed. J. Denier & Jorgen S. Frederiksen), vol. 6, pp. 415–430. World Scientific.CrossRefGoogle Scholar
Diamond, P., Champeaux, S., Malkov, M., Das, A., Gruzinov, I., Rosenbluth, M., Holland, C., Wecht, B., Smolyakov, A., Hinton, F., et al. 2001 Secondary instability in drift wave turbulence as a mechanism for zonal flow and avalanche formation. Nucl. Fusion 41 (8), 1067.CrossRefGoogle Scholar
Diamond, P., Liang, Y.-M., Carreras, B. & Terry, P. 1994 Self-regulating shear flow turbulence: a paradigm for the L to H transition. Phys. Rev. Lett. 72 (16), 2565.CrossRefGoogle Scholar
Dimits, A. M., Bateman, G., Beer, M., Cohen, B., Dorland, W., Hammett, G., Kim, C., Kinsey, J., Kotschenreuther, M., Kritz, A., et al. 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas 7 (3), 969–983.CrossRefGoogle Scholar
Hammett, G., Beer, M., Dorland, W., Cowley, S. & Smith, S. 1993 Developments in the gyrofluid approach to tokamak turbulence simulations. Plasma Phys. Control. Fusion 35 (8), 973.CrossRefGoogle Scholar
Hasegawa, A. & Mima, K. 1977 Stationary spectrum of strong turbulence in magnetized nonuniform plasma. Phys. Rev. Lett. 39 (4), 205.CrossRefGoogle Scholar
Ivanov, P. G., Schekochihin, A., Dorland, W., Field, A. & Parra, F. 2020 Zonally dominated dynamics and Dimits threshold in curvature-driven ITG turbulence. arXiv:2004.04047.Google Scholar
Kim, E.-J. & Diamond, P. 2002 Dynamics of zonal flow saturation in strong collisionless drift wave turbulence. Phys. Plasmas 9 (11), 4530–4539.CrossRefGoogle Scholar
Kim, E.-J. & Diamond, P. 2003 Zonal flows and transient dynamics of the $L$–$H$ transition. Phys. Rev. Lett. 90 (18), 185006.CrossRefGoogle ScholarPubMed
Kobayashi, S., Gürcan, Ö. D. & Diamond, P. H. 2015 Direct identification of predator-prey dynamics in gyrokinetic simulations. Phys. Plasmas 22 (9), 090702.CrossRefGoogle Scholar
Kobayashi, S. & Rogers, B. N. 2012 The quench rule, Dimits shift, and eigenmode localization by small-scale zonal flows. Phys. Plasmas 19 (1), 012315.CrossRefGoogle Scholar
Kolesnikov, R. A. & Krommes, J. 2005 Transition to collisionless ion-temperature-gradient-driven plasma turbulence: a dynamical systems approach. Phys. Rev. Lett. 94 (23), 235002.CrossRefGoogle ScholarPubMed
Krommes, J. A. & Kim, C.-B. 2000 Interactions of disparate scales in drift-wave turbulence. Phys. Rev. E 62 (6), 8508–8539.CrossRefGoogle ScholarPubMed
Lin, Z., Hahm, T. S., Lee, W., Tang, W. M. & White, R. B. 1998 Turbulent transport reduction by zonal flows: massively parallel simulations. Science 281 (5384), 1835–1837.CrossRefGoogle ScholarPubMed
Malkov, M., Diamond, P. & Smolyakov, A. 2001 On the stability of drift wave spectra with respect to zonal flow excitation. Phys. Plasmas 8 (5), 1553–1558.CrossRefGoogle Scholar
Mikkelsen, D. & Dorland, W. 2008 Dimits shift in realistic gyrokinetic plasma-turbulence simulations. Phys. Rev. Lett. 101 (13), 135003.CrossRefGoogle ScholarPubMed
Moyal, J. E. 1949 Quantum mechanics as a statistical theory. Math. Proc. Camb. Phil. Soc. 45 (1), 99–124.CrossRefGoogle Scholar
Numata, R., Ball, R. & Dewar, R. L. 2007 Bifurcation in electrostatic resistive drift wave turbulence. Phys. Plasmas 14 (10), 102312.CrossRefGoogle Scholar
Rath, F., Peeters, A., Buchholz, R., Grosshauser, S., Seiferling, F. & Weikl, A. 2018 On the tertiary instability formalism of zonal flows in magnetized plasmas. Phys. Plasmas 25 (5), 052102.CrossRefGoogle Scholar
Ricci, P., Rogers, B. N. & Dorland, W. 2006 Small-scale turbulence in a closed-field-line geometry. Phys. Rev. Lett. 97 (24), 245001.CrossRefGoogle Scholar
Rogers, B. & Dorland, W. 2005 Noncurvature-driven modes in a transport barrier. Phys. Plasmas 12 (6), 062511.CrossRefGoogle Scholar
Rogers, B., Dorland, W. & Kotschenreuther, M. 2000 Generation and stability of zonal flows in ion-temperature-gradient mode turbulence. Phys. Rev. Lett. 85 (25), 5336.CrossRefGoogle ScholarPubMed
Ruiz, D., Parker, J., Shi, E. & Dodin, I. 2016 Zonal-flow dynamics from a phase-space perspective. Phys. Plasmas 23 (12), 122304.CrossRefGoogle Scholar
Sakurai, J. J. 1994 Modern Quantum Mechanics Revised Edition. Addison–Wesley, edited by Tuan, S. F..Google Scholar
Smolyakov, A. & Diamond, P. 1999 Generalized action invariants for drift waves-zonal flow systems. Phys. Plasmas 6 (12), 4410–4413.CrossRefGoogle Scholar
St-Onge, D. A. 2017 On non-local energy transfer via zonal flow in the Dimits shift. J. Plasma Phys. 83 (5), 905830504.CrossRefGoogle Scholar
Tang, W. M. 1978 Microinstability theory in tokamaks. Nucl. Fusion 18 (8), 1089.CrossRefGoogle Scholar
Terry, P. & Horton, W. 1982 Stochasticity and the random phase approximation for three electron drift waves. Phys. Fluids 25 (3), 491–501.CrossRefGoogle Scholar
Terry, P. & Horton, W. 1983 Drift wave turbulence in a low-order $k$ space. Phys. Fluids 26 (1), 106–112.CrossRefGoogle Scholar
Weinbub, J. & Ferry, D. 2018 Recent advances in Wigner function approaches. Appl. Phys. Rev. 5 (4), 041104.CrossRefGoogle Scholar
Zhu, H., Zhou, Y. & Dodin, I. 2018 a On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. Phys. Plasmas 25 (8), 082121.CrossRefGoogle Scholar
Zhu, H., Zhou, Y. & Dodin, I. 2018 b On the structure of the drifton phase space and its relation to the Rayleigh–Kuo criterion of the zonal-flow stability. Phys. Plasmas 25 (7), 072121.CrossRefGoogle Scholar
Zhu, H., Zhou, Y. & Dodin, I. 2020 Theory of the tertiary instability and the Dimits shift from reduced drift-wave models. Phys. Rev. Lett. 124 (5), 055002.CrossRefGoogle ScholarPubMed
Zhu, H., Zhou, Y., Ruiz, D. & Dodin, I. 2018 c Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation. Phys. Rev. E 97 (5), 053210.CrossRefGoogle ScholarPubMed