Flux-driven turbulent transport using penalisation in the Hasegawa-Wakatani system

TL;DR

This work develops P-FLARE, a penalised, pseudo-spectral solver for flux-driven reduced-fluid models, and applies it to the flux-driven Hasegawa-Wakatani system. By using buffer-zone penalisation, it achieves efficient FFT-based simulations of radial boundaries while allowing the mean density gradient to evolve under turbulent transport. The results reveal density-profile relaxation with turbulent spreading, a subdiffusive turbulent front, and a transition to zonal-flow–dominated states that can suppress transport, along with sandpile-like self-organised bistability when a local particle source is introduced. The study demonstrates a flexible framework for exploring flux-driven turbulence, zonal flows, and L–H-like transitions in reduced-fluid models, with future extensions to self-consistent mean flows and broader reduced-model classes.

Abstract

First numerical results from the newly-developed pseudo-spectral code P-FLARE (Penalised FLux-driven Algorithm for REduced models) are presented. This flux-driven turbulence/transport code uses a pseudo-spectral formulation with the penalisation method in order to impose radial boundary conditions. Its concise, flexible structure allows implementing various quasi-two dimensional reduced fluid models in flux-driven formulation. Here, results from simulations of the modified Hasegawa-Wakatani system are discussed, where particle transport and zonal flow formation, together with profile relaxation, are studied. It is shown that coupled spreading/profile relaxation that one obtains for this system is consistent with a simple one dimensional model of coupled spreading/transport equations. Then, the effect of a particle source is investigated, which results in the observation of sandpile-like critical behaviour. The model displays profile stiffness for certain parameters, with very different input fluxes resulting in very similar mean density gradients. This is due to different zonal flow levels around the critical value for the control parameter (i.e. the ratio of the adiabaticity parameter to the mean gradient) and the existence for this system of a hysteresis loop for the transition from 2D turbulence to a zonal flow dominated state.

Figures (18)

• Figure 1: Decomposition of a given radial profile $n_{r}$ (blue) into its linear component $n_{lin}$ (dashed green) and its zonal fluctuating component $\overline{n}$ (dashdot red). Notice that using such a decomposition, we find that the zonal profile is periodic at the edges, while its derivative $\partial_{x}\overline{n}$ is not.
• Figure 2: Decomposition of the radial profile $n_{r}$ (blue) into its linear component $n_{lin}$ (dashed green) and its zonal fluctuating component $\overline{n}$ (dashdot red). The background density gradient is now computed between $x_{b1}$ and $x_{b2}$, and the buffer zones are indicated by the dashed lines.
• Figure 3: (a): smooth transition function $h$ defined according to Eq. \ref{['eq:smooth-transition']} on $[0,1]$. (b): smooth gate $\Psi$ defined from Eq. \ref{['eq:smooth-gate']} on $[0,L_{x}]$, where the transition parts are shown in orange.
• Figure 4: Top plot: modified zonal density perturbation $\overline{n}_{matched}$ (red), which is now periodic and flat at both ends. The corresponding matched radial profile $n_{r,matched}$ is shown in blue. The orignal radial profile $n_{r}$ (dashed lightblue), linear profile $n_{lin}$ (green) and zonal profile $\overline{n}$ (dashed darkred) from figure \ref{['fig:decomp-buffer']} are also shown. Bottom plot: smooth-gate function $\Psi$.
• Figure 5: Combined plot of $\psi_{mask}(x)=1-H(x)$ (blue) and the smooth-gate function $\Psi_{smooth}(x)$ (dashed red). Their radial extensions $\delta x_{b}$ and $\delta x_{m}$ are also shown (respectively blue and red arrows).