Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation

Abstract

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.

Nonlinear saturation and oscillations of collisionless zonal flows

“…(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.…”

“…Some progress in this area has been made by applying quasilinear (QL) models (section 2.4), such as the secondorder cumulant expansion theory (CE2) [9][10][11][12][13][14], or the stochastic structural stability theory [30][31][32]; however, those are not particularly intuitive. A more intuitive paradigm was proposed based on a simpler QL model known as the wave-kinetic equation (WKE) [6,7,[33][34][35][36][37][38][39]. The WKE treats DW turbulence as a collection of DW quanta ('driftons'), for which the ZF velocity serves as a collective field.…”

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

“…Among statistical QL theories, the wave kinetic equation (WKE) is a popular model that captures the essential basic physics of DW turbulence, e.g. the formation of ZFs (Parker 2016;Ruiz et al 2016;Zhu et al 2018c;Parker 2018;Zhu et al 2018a,b;Diamond et al 2005;Smolyakov & Diamond 1999;Smolyakov et al 2000; † Email address for correspondence: deruiz@sandia.gov arXiv:1901.02518v1 [physics.plasm-ph] 8 Jan 2019…”

“…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.)…”

“…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):…”