Abstract
We review some recent results that have been obtained in the investigation of zonal flow emergence, by means of a gyrokinetic trapped ion model, in the regime of ion temperature gradient instabilities for tokamak plasmas. We show that an analogous formulation of the zonal flow dynamics in terms of the Reynolds tensor applies in the fluid and kinetic regimes, where polarization effects play a major role. The kinetic regime leads to the emergence of a resonant mode at a frequency close to the drift frequency. With the objective of modeling both separate fluid and kinetic regimes of zonal flows, we used in this paper a methodology for deriving both Charney–Hasegawa–Mima (CHM) and Hasegawa–Wakatani models. This methodology is based on the trapped ion model and is analogous to the hierarchy leading from the Vlasov equation to the macroscopic fluid equations. The nature of zonal flows in the hierarchy of the Mima, Hasegawa and Wakatani models is investigated and discussed through comparisons with global kinetic simulations. Applications to the CHM equation are discussed, which applies to a broad variety of hydrodynamical systems, ranging from large-scale processes met in magnetically confined plasma to the so-called zonostrophy turbulence emerging in the case of small-scale forced, two-dimensional barotropic turbulence (Sukoriansky et al. Phys. Rev. Letters, 101, 178501, 2008).
Highlights
Understanding and controlling turbulence is one of the key elements for a successful magnetic confinement of a tokamak plasma
We have shown that the zonal flow generation in collisionless plasmas is controlled by the Reynolds tensor, usually recognized as the main actor in the fluid approach, but explicitly depends on the microscopic physics that govern the ion temperature gradient instability
We have found that the Hasegawa–Wakatani model can be recovered from the gyrokinetic trapped ion model when interchange-type turbulence is considered, and that the zonal flow dynamics is strongly impeded by polarization effects
Summary
Understanding and controlling turbulence is one of the key elements for a successful magnetic confinement of a tokamak plasma. [26], associated with a low frequency signal at a few kilohertz, i.e., at a much lower frequency than the usual geodesic acoustic mode, the geodesic version of ion acoustic modes Central to all these physical turbulent systems, be they in the core region of the tokamak plasma or in the shallow rotating atmosphere, is the generation of ZFs, exhibiting low frequency oscillations, which are believed to be responsible for suppressing small-scale turbulence and stabilizing the turbulence. The main interest in the HW and MHW models lays in their debated capability to account for a self-consistent generation of zonal flows by small-scale turbulence, and in the possibility to shed light in the spontaneous transition to a turbulence-suppressed regime In this respect, these models display some differences. Labeling with “(U ) ”, the time derivatives of the energy and enstrophy in this unmodified HW model, they obey dE (U )
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