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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: hzhu@pppl.gov
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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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