Theory of the Tertiary Instability and the Dimits Shift from Reduced Drift-Wave Models

Hongxuan Zhu, Yao Zhou, and I. Y. Dodin

Phys. Rev. Lett. 124, 055002 – Published 4 February 2020

Abstract

Tertiary modes in electrostatic drift-wave turbulence are localized near extrema of the zonal velocity $U (x)$ with respect to the radial coordinate $x$ . We argue that these modes can be described as quantum harmonic oscillators with complex frequencies, so their spectrum can be readily calculated. The corresponding growth rate $γ_{TI}$ is derived within the modified Hasegawa-Wakatani model. We show that $γ_{TI}$ equals the primary-instability growth rate plus a term that depends on the local $U^{''}$ ; hence, the instability threshold is shifted compared to that in homogeneous turbulence. This provides a generic explanation of the well-known yet elusive Dimits shift, which we find explicitly in the Terry-Horton limit. Linearly unstable tertiary modes either saturate due to the evolution of the zonal density or generate radially propagating structures when the shear $| U^{'} |$ is sufficiently weakened by viscosity. The Dimits regime ends when such structures are generated continuously.

Received 13 October 2019
Revised 11 December 2019
Accepted 14 January 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.055002

Physics Subject Headings (PhySH)

Drift waves Flow instability Plasma transport Shear flows Transition to turbulence

Plasma Physics

Authors & Affiliations

Hongxuan Zhu ^1,2,*, Yao Zhou ¹, and I. Y. Dodin ^1,2

¹Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA
²Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA

^*hzhu@pppl.gov

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Vol. 124, Iss. 5 — 7 February 2020

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Images

Figure 1
Results of MHWE simulations using dedalus [23] with $α = 5$ , $κ = 12$ , and $\hat{D} = 0.1 \nabla^{4}$ . The system is initialized with small random noise. (a) A snapshot at $t = 1000$ showing fluctuations $\tilde{w}$ (color), $U$ (solid line), and $κ_{eff} ≐ κ - N^{'}$ (dashed line). (b) Evolution of the DW energy $E_{DW} ≐ \int ({\tilde{n}}^{2} + | \nabla \tilde{φ} |^{2}) d x / 2$ , and the zonal energy $E_{U} = \int U^{2} d x / 2$ and $E_{N} = \int N^{2} d x / 2$ . The inset is an enlargement showing the energies in log scale at $0 < t < 50$ when the primary and secondary instabilities develop. Type-I PP oscillations are indicated by blue circles, and a type-II PP oscillation is indicated by the red star. (See text for more details.)
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Figure 2
(a) Trapped and runaway modes (plotted together) for the zonal profile (6): Numerical solutions of Eq. (4) (color) versus analytic solutions (9) (dashed contours) at $α = 5$ , $κ = 12$ , $q_{Z} = 0.3$ , $u = 10$ , and $p_{y} = 1$ . The green curve shows $U$ . (b) Same mode structures in phase space: Wigner functions (color) versus isosurfaces of the Hamiltonian (7) (dashed contours). (c) Numerical (labeled “ $N$ ”) and analytic (labeled “ $A$ ”) growth rates versus $u$ . The subscripts “ $T$ ” and “ $R$ ” stand for trapped and runaway, respectively. The analytic growth rates are obtained by solving Eq. (9) iteratively with $ω = p_{y} U_{0} + p_{y} κ / {\bar{p}}_{0}^{2} - i D_{0}$ as the initial guess. (d) Dimits shift in the Terry-Horton limit: Simulation results of the modified Terry-Horton equation with $δ_{p} = δ_{0} p_{y}$ and $\hat{D} = 1 - 0.01 \nabla^{2}$ (green circles and red crosses) versus analytic results [Eq. (13) with $p_{y} = 1$ and $ϱ = 0.05$ ]. Simulation details are the same as in Ref. [15] (cf. Fig. 7 therein).
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Figure 3
(a) The growth rates of the trapped ( $γ_{T}$ , circles) and runaway ( $γ_{R}$ , triangles) modes found numerically for the zonal profile (14) at $q_{Z} = 0.3$ and $u = 20$ . Different colors indicate different values of $N_{0}$ for trapped (runaway) modes: blue, $N_{0} = 0$ ( $N_{0} = 0$ ); red, $N_{0} = 5.4$ ( $N_{0} = 1.8$ ); green, $N_{0} = 9$ ( $N_{0} = 3.6$ ). The peak of $γ_{T}$ shifts to smaller $p_{y}$ , but the peak of $γ_{R}$ remains at $p_{y} \approx 1.2$ . (b) The critical shear $S_{c} = q_{Z} u_{c}$ versus $κ$ at $α = 5$ , $q_{Z} = 0.3$ , and $p_{y} = 0.4$ . We also plot $γ_{PI}$ , and $γ_{T}$ at $u = u_{c}$ (denoted $γ_{T, c}$ ).
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Figure 4
Snapshots of a propagating structure corresponding to (a) $t = 756$ , (b) $t = 763$ , (c) $t = 770$ , and (d) $t = 777$ of Fig. 1. The axes are rotated by 90° compared with Fig. 1. This structure originates as a trapped mode ( $p_{y} = 0.4$ ) at $x \approx - 10$ and propagates to the neighboring minimum of $U$ at $x \approx - 20$ . Both the minimum and the maximum of $U$ are amplified by this structure, resulting in an increase of $E_{U}$ .
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