Description
The Hasegawa-Wakatani (HW) system is the minimal non-trivial model for instability-driven tokamak turbulence, which exhibits the formation of zonal flows (ZFs) as an example of self-organisation. Particularly, it displays a phase transition between 2D turbulence and a quasi-1D ZF dominated state, as its linear parameters are varied, with a hysteresis loop around the transition point. It thus constitutes a convenient test bench in order to design reduced models for turbulent transport, that include these complex phenomena. For instance, the transition observed in direct numerical simulations (DNS) can be reproduced using a low order wavenumber space network model composed of only 12 Fourier modes.
A more realistic reduction, inspired by the generalised quasi-linear approximation and the Tokam1D model, called poloidally truncated models (PTMs), is proposed. Such reduction consists in keeping the complete resolution in the radial direction, while retaining only a few poloidal modes. PTMs hence have the minimal ingredients for linear instability, non-linear turbulent interactions, and ZF formation, while describing completely resolved profiles, and run nearly 20 times as fast as DNS.
In the fixed-gradient HW model, it is found that at least 4 modes, distributed around the most unstable mode, are needed to reproduce reasonably the transition between turbulence and ZFs observed in DNS, although the transition point is systematically shifted. Using these reductions, the particle flux can be correctly reproduced, except in the vicinity of the transition point.
Finally, applying PTMs to the flux-driven HW system allows to quantitatively match the ZF level and the time evolution of the mean density gradient observed in DNS, in both turbulent and ZF dominated regime. Furthermore, using 10 poloidal modes, the measured particle flux distribution function yields very good agreement with that observed in DNS, meaning that PTMs reproduce correctly both sationnary state and its statistical properties.