Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conventional quantum particles. This causes unusual phase-space dynamics that is typically not captured by the WKE. We demonstrate how to correctly model this dynamics with the WME instead. Specifically, we report full-wave phase-space simulations of the zonal-flow formation (zonostrophic instability), deterioration (tertiary instability), and the so-called predator-prey oscillations. We also show how the WME facilitates analysis of these phenomena, namely, (i) we show that full-wave effects critically affect the zonostrophic instability, particularly its nonlinear stage and saturation; (ii) we derive the tertiary-instability growth rate; and (iii) we demonstrate that, with full-wave effects retained, the predator-prey oscillations do not require zonal-flow collisional damping, contrary to previous studies. We also show how the famous Rayleigh-Kuo criterion, which has been missing in wave-kinetic theories of drift-wave turbulence, emerges from the WME.

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...

The phase space of driftons (drift-wave quanta) is studied within the generalized Hasegawa-Mima collisionless-plasma model in the presence of zonal flows. This phase space is made intricate by the corrections to the drifton ray equations that were recently proposed by Parker [J. Plasma Phys. 82, 595820602 (2016)] and Ruiz et al. [Phys. Plasmas 23, 122304 (2016)]. Contrary to the traditional geometrical-optics (GO) model of the drifton dynamics, it is found that driftons can be not only trapped or passing, but they can also accumulate spatially while experiencing indefinite growth of their momenta. In particular, it is found that the Rayleigh-Kuo threshold known from geophysics corresponds to the regime when such "runaway" trajectories are the only ones possible. On one hand, this analysis helps visualize the development of the zonostrophic instability, particularly its nonlinear stage, which is studied here both analytically and through wave-kinetic simulations. On the other hand, the GO theory predicts that zonal flows above the Rayleigh-Kuo threshold can only grow; hence, the deterioration of intense zonal flows cannot be captured within a GO model. In particular, this means that the so-called tertiary instability of intense zonal flows cannot be adequately described within the quasilinear wave kinetic equation, contrary to some previous studies.