On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows

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

doi:10.1063/1.5038859

Cited by 20 publications

(37 citation statements)

References 48 publications

(118 reference statements)

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“…Our derivation makes use of the Weyl calculus, in conjunction with the quasinormal statistical closure and the well-known geometrical-optics assumptions. The obtained model is similar to previous reported works (Parker 2018;Ruiz et al 2016;Zhu et al 2018b;Parker 2016;Zhu et al 2018a) but also includes a nonlinear term describing wave-wave scattering. Unlike in WTT, the collision operator depends on the local ZF velocity and breaks down at the Rayleigh-Kuo threshold.…”

Section: Discussionsupporting

confidence: 87%

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

2019

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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 literature. Singh et al. 2014). The WKE has the intuitive form of the Liouville equation for the DW action density J in the ray phase space (Parker 2016;Ruiz et al. 2016;Zhu et al. 2018c;Parker 2018; Zhu et al. 2018a,b):where Ω is the local DW frequency, Γ is a dissipation rate due to interactions with ZFs, and {·, ·} is the canonical Poisson bracket. (For the sake of clarity, terms related to external forcing and dissipation are omitted here.) However, as with all QL models, Eq. (1.1) neglects nonlinear wave-wave scattering, and in consequence, is not able to capture the Batchelor-Kraichnan inverse-energy cascade (Srinivasan & Young 2012) or produce the Kolmogorov-Zakharov spectra for DWs (Connaughton et al. 2015). Hence, a question remains as to whether the existing WKE for inhomogeneous turbulence can be complemented with a wave-wave collision operator C[J, J]. The goal of this work is to calculate C[J, J] explicitly. Starting from the generalized Hasegawa-Mima equations (gHME) (Krommes & Kim 2000; Smolyakov & Diamond 1999), we derive a WKE with a DW collision operator C[J, J] for inhomogeneous DW turbulence. Our derivation is based on the Weyl calculus (Weyl 1931), which makes our approach similar to that in Ruiz et al. (2016) where the QL approximation was used. However, in contrast to Ruiz et al. (2016), we do not rely on the QL approximation here but instead account for DW collisions perturbatively using the quasinormal approximation. The main result of this work are Eqs. (4.12) and (4.22). In this final result, DWs are modeled in a similar manner as in Eq. (1.1). The difference is that Eq. (4.12) includes nonlinear DW scattering, which is described by a wave-wave collision operator C[J, J] that is bilinear in the DW wave-action density J. The resulting model conserves the two nonlinear invariants of the gHME, which are the total enstrophy and the total energy. The present formulation is fundamentally different from the previously reported homogeneous weak-wave-turbulence mode...

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Section: Discussionsupporting

confidence: 87%

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

2019

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“…(Note that this estimate reinstates the dependence on q, which is absent in M.) Second, the RK criterion does not describe the ZF saturation but rather determines the threshold of the instability of the Kelvin-Helmholtz type that destroys the ZF. (It is also called the 'tertiary instability' by some authors [3,35,36,[55][56][57][58][59], and in our earlier studies we showed that this instability does not exist in the GO limit [35,36,55].) Since the RK threshold corresponds to u>u c,2 (section 3.2), and u u c c ,2 ,1  in the GO regime, we claim that ZFs saturate before the RK threshold is reached.…”

supporting

confidence: 54%

“…They also illustrate the dynamics of test driftons, which are added post hoc. Their trajectories are calculated using the ray equations [55], where the drifton Hamiltonian  is determined by the ZF velocity U that is shown in the lower row and, for test-drifton simulations, treated as a prescribed function of (y, t). In figure 9(a), most driftons are passing and trapped, which leads to ZF oscillations.…”

Section: Numerical Simulations 41 Parameter Scanmentioning

confidence: 99%

Nonlinear saturation and oscillations of collisionless zonal flows

2019

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In homogeneous drift-wave turbulence, zonal flows (ZFs) can be generated via a modulational instability (MI) that either saturates monotonically or leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is often attributed as the predator-prey oscillations induced by ZF collisional damping; however, similar dynamics is also observed in collisionless ZFs, in which case a different mechanism must be involved. Here, we propose a semi-analytic theory that explains the transition between the oscillations and saturation of collisionless ZFs within the quasilinear Hasegawa-Mima model. By analyzing phase-space trajectories of DW quanta (driftons) within the geometrical-optics (GO) approximation, we argue that the parameter that controls this transition is N∼γ MI /ω DW , where γ MI is the MI growth rate and ω DW is the linear DW frequency. We argue that at N=1, ZFs oscillate due to the presence of so-called passing drifton trajectories, and we derive an approximate formula for the ZF amplitude as a function of time in this regime. We also show that at N 1, the passing trajectories vanish and ZFs saturate monotonically, which can be attributed to phase mixing of higher-order sidebands. A modification of N that accounts for effects beyond the GO limit is also proposed. These analytic results are tested against both quasilinear and fully-nonlinear simulations. They also explain the earlier numerical results by Connaughton et al (2010 J. Fluid Mech. 654 207) and Gallagher et al (2012 Phys. Plasmas 19 122115) and offer a different perspective on what the control parameter actually is that determines the transition from the oscillations to saturation of collisionless ZFs.

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“…As shown in Ref. [33], for α → ∞, the KHI is suppressed at all q 2 Z < 1. Here, α is finite but still large (α = 5), so similar conclusions apply.…”

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confidence: 52%

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

2020

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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 γTI is derived within the modified Hasegawa-Wakatani model. We show that γTI 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.Drift-wave (DW) turbulence plays a significant role in fusion plasmas and can develop from various "primary" instabilities [1][2][3][4][5][6][7]. However, having their linear growth rate γ PI above zero is not enough to make plasma turbulent, because the "secondary" instability can suppress turbulence by generating zonal flows (ZFs) [5][6][7][8][9][10][11][12]; hence, the threshold for the onset of turbulence is modified compared to the linear theory. This constitutes the so-called Dimits shift [12][13][14][15][16][17], which has been attracting attention for two decades. The finite value of the Dimits shift is commonly attributed to the "tertiary" instability (TI) [5], and some theories of the TI have been proposed [5,[15][16][17][18][19][20]. However, basic understanding and generic description of the TI and the Dimits shift have been elusive.Here, we propose a simple yet quantitative theory of the TI using the modified Hasegawa-Wakatani equation (mHWE) [2,16] as a base turbulence model. We clarify several misconceptions regarding the TI, and we explicitly derive the Dimits shift in the limit corresponding to the Terry-Horton model [15,21]. Our approach is also applicable to other DW models, such as ion-temperaturegradient (ITG) turbulence [5], as discussed towards the end. Furthermore, we explain TI's role in two types of predator-prey (PP) oscillations, in determining the characteristic ZF scale in the Dimits regime, and in transition to the turbulent state.Model equations.-The mHWE [2, 16] is a slab model of two-dimensional electrostatic turbulence with a uniform magnetic field B = Bẑ. Turbulence is considered on the plane (x, y), where x is the radial coordinate and y is the poloidal coordinate. The model describes ϕ and n, which are fluctuations of the electric potential and density, respectively. Ions are assumed cold while electrons have finite temperature T e . The plasma is assumed to have an equilibrium density profile n 0 (x) parameterized by a constant κ . = a/L n , where a is the system length and L n . = (−n 0 /n 0 ) −1 . (We use . = to denote...

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On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows

Cited by 20 publications

References 48 publications

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

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

Nonlinear saturation and oscillations of collisionless zonal flows

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

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