Hasegawa-Wakatani Model Research Articles

Anomalous transport of multi-species edge plasma with the generalized Hasegawa–Wakatani model and the FLR effects

Anomalous transport of multi-species edge plasma with the generalized Hasegawa–Wakatani model and the FLR effects

StyleGAN as an AI deconvolution operator for large eddy simulations of turbulent plasma equations in BOUT++

We present the use of StyleGAN, a face-synthesis generative adversarial network (GAN) developed by NVidia, as a deconvolution operator for large eddy simulation (LES) of plasma turbulence. The overall methodology, named style eddy simulation, has been integrated into the BOUT++ solver and tested on the original and modified Hasegawa–Wakatani models using different mesh sizes, 2562 and 5122, and different values of the adiabaticity parameter α and background density gradient κ. Using a LES resolution of 32 × 32 and 64 × 64, i.e., 64× smaller resolution than the corresponding direct numerical simulation (DNS), results show convergence toward the ground truth as we tighten the reconstruction tolerance, and an algorithm complexity O(N log N) is compared to the O(N2) of BOUT++. Finally, the trained GAN can be used to create valid initial conditions for a faster DNS by avoiding to start from nonphysical initial perturbations.

Resistive drift wave turbulence and anomalous transport of multi-species plasma

Anomalous transport of multi-species plasma is considered with the generalized Hasegawa–Wakatani model. It is shown that the transport of all plasma species is described by fractional diffusion equations with the same effective diffusion coefficient. Strongly enhanced perturbations of heavy impurity density are found in long-living plasma flow vortices.

HW2D: A reference implementation of the Hasegawa-Wakatani model for plasma turbulence in fusion reactors

HW2D: A reference implementation of the Hasegawa-Wakatani model for plasma turbulence in fusion reactors

TIFF: Gyrofluid turbulence in full-f and full-k

A model and code (“TIFF”) for isothermal gyrofluid computation of quasi-two-dimensional interchange and drift wave turbulence in magnetized plasmas with arbitrary fluctuation amplitudes (“full-f”) and arbitrary polarization wavelengths (“full-k”) is introduced. The model reduces to the paradigmatic Hasegawa-Wakatani model in the limits of small turbulence amplitudes (“delta-f”), cold ions (without finite Larmor radius effects), and homogeneous magnetic field. Several solvers are compared for the generalized Poisson problem, that is intrinsic to the full-f gyrofluid (and gyrokinetic) polarization equation, and a novel implementation based on a dynamically corrected Fourier method is proposed. The code serves as a reference case for further development of three-dimensional full-f full-k models and solvers, and for fundamental exploration of large amplitude turbulence in the edge of magnetized plasmas.

On the Effects of Tokamak Plasma Edge Symmetries on Turbulence Relaxation

The plasma edge of a tokamak configuration is characterized by turbulent dynamics leading to enhanced transport. We construct a simplified 3D Hasegawa–Wakatani model reducing to a single partial differential equation for the turbulent electric potential dynamics. Simulations demonstrate how the 3D turbulence relaxes on a 2D axisymmetric profile, corresponding to the so-called interchange turbulence. The spectral features of this regime are found to be strongly dependent on the initialization pattern. We outline that the emergence of axisymmetric turbulence is also achieved when the corresponding mode amplitude is not initialized. Then, we introduce the symmetries of the magnetic X-point of a tokamak configuration. We linearize the governing equation by treating the poloidal field as a small correction. We show that it is not always possible to solve the electric potential dynamics following a perturbative approach. This finding, which is due to resonance between the modes of the background and the poloidal perturbation, confirms that the X-point symmetries can alter the properties of turbulent transport in the edge region.

On the 3D turbulence regime in a Tokamak plasma edge

We derive a reduced model for the electrostatic turbulence in a Tokamak edge, when dealing with a resistive plasma and neglecting the spatial gradient of the background density which triggers the linear drift wave response. The obtained dynamics, de facto equivalent to a Hasegawa–Wakatani model, is characterized by a constitutive relation between the electric potential Laplacian and the density, which allows to deal with a single 3D equation governing the electric potential fluctuations. We study the evolution of the model, by separating the n=0 (n indicating toroidal-like number) mode from all the other ones. Then, we linearize the dynamics of the n≠0 modes around the steady 2D spectrum, which describes the spectral features of the electrostatic 2D turbulence. We theoretically and numerically demonstrate the existence of decaying branch of the 3D turbulence, having such a 2D steady spectrum as a natural attractor. This result suggests that the basic constituent of the self-sustained non-linear drift response in the plasma edge has to be individualized in the non-linear 2D electrostatic turbulence.

Data-driven model discovery for plasma turbulence modelling

An important problem in nuclear fusion plasmas is the prediction and control of turbulence which drives the cross-field transport, thus leading to energy loss from the system and deteriorating confinement. Turbulence, being a highly nonlinear and multiscale process, is challenging to theoretically describe and computationally model. Most advanced computational models fall into one of the two categories: fluid or gyro-kinetic. They both come at a high computational cost and cannot be applied for routine simulation of plasma discharge evolution and control. Development of reduced models based on (physics informed) artificial neural networks could potentially fulfil the need for affordable simulations of plasma turbulence. However, the training requires an extensive data base and the obtained models lack extrapolation capability to scenarios not originally encountered during training. This leads to reduced models of limited validity which may not prove adequate for predicting scenarios in future machines. In contrast, we explore a data-driven model discovery approach based on sparse regression to infer governing nonlinear partial differential equations directly from the data. Our input data are generated by simulations of drift-wave turbulence according to the Hasegawa–Wakatani and modified Hasegawa–Wakatani models. Balancing model accuracy and complexity enables the reconstruction of the systems of partial differential equations accurately describing the dynamics simulated in the input data sets. Sparse regression is not data hungry and can be extrapolated to unexplored parameter ranges. We explore and demonstrate the potential of this approach for fusion plasma turbulence modelling. The findings show that the methodology is promising for the development of reduced and computationally efficient turbulence models as well as for existing model cross-validation.

Lagrangian conditional statistics and flow topology in edge plasma turbulence

Lagrangian statistics and particle transport in edge plasma turbulence are investigated using the Hasegawa–Wakatani model and its modified version. The latter shows the emergence of pronounced zonal flows. Different values of the adiabaticity parameter are considered. The main goal is to characterize the role of coherent structures, i.e., vortices and zonal flows, and their impact on the Lagrangian statistics of particles. Computationally intensive long time simulations following ensembles of test particles over hundreds of eddy turnover times are considered in statistically stationary turbulent flows. The flow topology is characterized using the Lagrangian Okubo–Weiss criterion in order to split the flow into topologically different domains. In elliptic and hyperbolic regions, the probability density functions (PDFs) of the residence time have self-similar algebraic decaying tails. However, in the intermediate regions, the PDFs exhibit exponentially decaying tails. Topologically conditioned PDFs of the Lagrangian velocity, and acceleration and density fluctuations are likewise computed. The differences between the classical Hasegawa–Wakatani system and its modified version are assessed, and the role of zonal flows is highlighted. The density flux spectrum, which characterizes the contributions of different length scales, is studied, and its inertial scaling is found to be in agreement with predictions based on dimensional arguments. Analyzing the angular change of particle tracers at different time scales, corresponding to coarse grained curvature, completes the study, and these multiscale geometric statistics quantify the directional properties of the particle motion in different flow regimes.

Consistent coupling algorithms for coupled core-edge simulations of plasma turbulence

Two schemes for coupling gyrokinetic simulations of microturbulence in tokamaks are proposed. The first scheme is based on an additive Schwarz domain decomposition. We show that, because the goal of turbulence is long-time averages of the dynamics rather than temporal accuracy, the iteration to self-consistency across domains, which is typically required by Schwarz schemes, can be avoided, thereby accelerating the computation. Second, we propose a coupling scheme that relies entirely on the addition of source terms, leaving the boundary conditions arbitrary. The practical motivations for such a scheme are discussed, and forms of the source terms that ensure consistency and stability are derived. The schemes are tested on a nonlinear, one-dimensional model problem, and the first scheme is further tested on the Hasegawa–Wakatani model.

Influence of zonal flow and density on resistive drift wave turbulent transport

The generations of zonal flow (ZF) and density (ZD) and their feedback on the resistive drift wave turbulent transport are investigated within the modified Hasegawa-Wakatani model. With proper normalization, the system is only controlled by an effective adiabatic parameter, α, where the ZF dominates the collisional drift wave (DW) turbulence in the adiabatic limit α>1. By conducting direct numerical simulations, we found that the ZF can significantly reduce the transport by trapping the DWs in the vicinities of its extrema for α>1, whereas the ZD itself has little impact on the turbulence but can only assist ZF to further decrease the transport by flattening the local plasma density gradient.

Dimits shift, avalanche-like bursts, and solitary propagating structures in the two-field flux-balanced Hasegawa–Wakatani model for plasma edge turbulence

We show that the recently introduced two-field flux-balanced Hasegawa–Wakatani (BHW) model captures the key features of drift-wave turbulent transport mediated by zonal flows observed in more complete and accurate gyrokinetic simulations, such as the existence of a nonlinear upshift of the threshold for drift wave turbulence driven transport, often called the Dimits shift, as well as non-local transport with avalanche bursts and solitary propagating structures. Because of the approximations made in the BHW model, these observations are made for the particle flux instead of the heat flux more commonly studied in ion temperature gradient (ITG) driven turbulence in fluid or gyrokinetic codes. Many of these features are not seen in other Hasegawa–Wakatani models, which confirm the critical role of the electron dynamics parallel to the magnetic field lines. To address questions regarding the role of boundary conditions on the drift-wave zonal flow dynamics, we apply our model to both a channel domain geometry and the more typical doubly periodic geometry. We only observe strong soliton-like solutions in the particle flux for the channel geometry, in the vicinity of the boundaries, where strong velocity shear and density gradients are generated, which are absent in the doubly periodic simulations. Changing the aspect ratio of the simulation domain also has a significant effect. In domains which are elongated in the radial direction, more complex multi-scale dynamics takes place, with multiple zonal jets interacting with each other, and large scale avalanches.

The influence of magnetic field curvature on intermittency in drift-wave turbulence in the stellarator TJ-K

The influence of magnetic field curvature on the intermittency in density and potential fluctuations in drift-wave turbulence has been investigated in the stellarator TJ-K in the framework of an extended Hasegawa–Wakatani model. A structure function analysis was used to estimate the intermittency level of poloidally resolved drift-wave turbulence measurements of a deuterium plasma in TJ-K. Potential fluctuations were found to be broadly self-similar, whereas density fluctuations were found to be more intermittent in the region with negative normal and positive geodesic curvatures. This behavior could be understood by comparing the data to two-dimensional extended Hasegawa–Wakatani simulations that retain gradients in the magnetic field strength, giving rise to curvature effects. The model is able to reproduce the trends in the experimental data if both normal and geodesic curvature effects are accounted for, as well as the local anisotropy of turbulent length scales. The analysis indicates the importance of local magnetic geometry as a factor in the decoupling of density and potential fluctuations, leading to intermittency in drift-wave turbulence.

Flux-balanced two-field plasma edge turbulence in a channel geometry

We investigate the drift wave–zonal flow interaction formulated on a channel domain geometry approximating an isolated plasma edge region with zero net radial transport across the boundary. The recent two-field flux-balanced Hasegawa–Wakatani (BHW) model with improved treatment for a parallel electron response is adapted to the channel geometry configuration, which allows for generalized non-uniform zonal density profiles and is valid for the simulation of turbulent fields near the tokamak or stellarator edge regions. New conserved quantities are constructed based on the channel geometry to help the analysis for the competition between zonal states and non-zonal fluctuations. Effective bounds can be found constraining the maximum growth of total fluctuations and the amplitude of the dominant zonal state based on the conserved quantities. Total statistical variance among all the modes can also be estimated depending on the zonal state strength. The theoretical discoveries are confirmed by detailed numerical experiments from simulations in the channel domain. In addition, the channel geometry provides further support for the important advantage of adopting the balanced flux correction in the BHW model by showing a physically consistent growth rate from a stability analysis for the small-amplitude fluctuation interaction with a prescribed zonal mean profile, in comparison with the persistent instability and strong outward transport found in the modified Hasegawa–Wakatani model even with the increasing zonal density profile. This is again confirmed by direct numerical simulations of the two models. The channel domain BHW model framework with attractive features implies many potential applications in the study of the complex phenomena in plasma edge turbulence.

A flux-balanced fluid model for collisional plasma edge turbulence: Numerical simulations with different aspect ratios

We investigate the drift wave–zonal flow dynamics in a shearless slab geometry with the new flux-balanced Hasegawa-Wakatani model. As in previous Hasegawa-Wakatani models, we observe a sharp transition from a turbulence dominated regime to a zonal jet dominated regime as we decrease the plasma resistivity. However, unlike previous models, zonal structures are always present in the flux-balanced model, even for high resistivity, and strongly reduce the level of particles and vorticity flux. The more robust zonal jets also have a higher variability than in previous models, which is further enhanced when the computational domain is chosen to be elongated in the radial direction. In these cases, we observe complex multiscale dynamics, with multiple jets interacting with one another, and intermittent bursts. We present a detailed statistical analysis which highlights how the changes in the aspect ratio of the computational domain affect the third-order statistical moments, and thus modify the turbulent dynamics. By changing the aspect ratio and extending either the radial or the binormal direction, the present model study offers a better approximation to mimic turbulence in flux tube simulations for either very low magnetic shear or very high magnetic shear systems.

Strong turbulence, self-organization and plasma confinement

This paper elucidates the close connections between hydrodynamic models of two-dimensional fluids and reduced models of plasma dynamics in the presence of a strong magnetic field. The key element is the similarity of the Coriolis force to the Lorentz force. The reduced plasma model, the Hasegawa–Mima equation, is equivalent to the two-dimensional ion vortex equation. The paper discusses the history of the Hasegawa–Mima model and that of a related reduced system called the Hasegawa–Wakatani model. The 2D fluid ↔ magnetized plasma analogy is exploited to argue that magnetized plasma turbulence exhibits a dual cascade, including an inverse cascade of energy. Generation of ordered mesoscopic flows in plasmas (akin to zonal jets) is also explained. The paper concludes with a brief explanation of the relevance of the quasi-2D dynamics to aspects of plasma confinement physics.

A flux-balanced fluid model for collisional plasma edge turbulence: Model derivation and basic physical features

We propose a new reduced fluid model for the study of the drift wave–zonal flow dynamics in magnetically confined plasmas. Our model can be viewed as an extension of the classic Hasegawa-Wakatani (HW) model and is based on an improved treatment of the electron dynamics parallel to the field lines, to guarantee a balanced electron flux on the magnetic surfaces. Our flux-balanced HW (bHW) model contains the same drift-wave instability as previous HW models, but unlike these models, it converges exactly to the modified Hasegawa-Mima model in the collisionless limit. We rely on direct numerical simulations to illustrate some of the key features of the bHW model, such as the enhanced variability in the turbulent fluctuations and the existence of stronger and more turbulent zonal jets than the jets observed in other HW models, especially for high plasma resistivity. Our simulations also highlight the crucial role of the feedback of the third-order statistical moments in achieving a statistical equilibrium with strong zonal structures. Finally, we investigate the changes in the observed dynamics when more general dissipation effects are included and, in particular, when we include the reduced model for ion Landau damping originally proposed by Wakatani and Hasegawa.

Localization of the eigenmode of the drift-resistive plasma by zonal flow

The mode structure of the linearly growing mode in the resistive-drift plasma is studied by the Hasegawa-Wakatani model with a sinusoidally forced zonal flow V. The parallel compression by the resistive current and the particle transport may drive the perturbations unstable. The linearly growing eigenstate characteristically maintains a positive phase difference between the electric potential and the generalized potential vorticity. The profile of the most unstable mode exhibits an asymmetry regarding the zonal flow: It is found that the mode is localized around where V is fast, thereby large curvature of V, in the direction of the electron diamagnetic drift.

Drift-wave observation in a toroidal magnetized plasma and comparison with a modified Hasegawa-Wakatani model

This paper presents the results of fluctuation measurements in a toroidal magnetized plasma, using Langmuir probes, and comparisons between the observed frequency modes and a Hasegawa-Wakatani model including curvature, adapted to the specifics of the toroidal device. More precisely, two kinds of signals are detected in the presence of an additional vertical magnetic field in the region of significant density and potential gradients. A high frequency, propagating component, corresponding to dissipative drift-waves in the curved magnetic field, is observed and the frequency and typical wavelengths are found to be in good agreement with the linear Hasegawa-Wakatani model including curvature effects. A second, low frequency component is observed at lower frequencies and is shown to correspond to large scale vertical electrostatic field structures. A significantly high level of cross correlation is observed between these two signals, with an identifiable time delay, which suggests an analogy to the time delayed quasi-periodic dynamics in predator-prey systems, and a similar phenomenon is observed between zonal flows and microturbulence in tokamaks and other magnetised plasma systems.

The role of zonal flows in reactive fluid closures

We will give an overview of results obtained by our reactive fluid model. It is characterised as a fluid model where all moments with sources in the experiment are kept. Furthermore, full account is taken for the highest moments appearing in unexpanded denominators also including full toroidicity. It has been demonstrated that the strength of zonal flows is dramatically larger in reactive fluid closures than in those which involve dissipation. This gives a direct connection between the fluid closure and the level of excitation of turbulence. This is because zonal flows are needed to absorb the inverse cascade in quasi 2D turbulence. This also explains the similarity in structure of the transport coefficients in our model with a reactive closure in the energy equation and models which have a reactive closure because of zero ion temperature such as the Hasegawa–Wakatani model. Our exact reactive closure unifies several well-known features of tokamak experiments such as the L–H transition, internal transport barriers and the nonlinear Dimits upshift of the critical gradient for onset of transport. It also gives transport of the same level as that in nonlinear gyrokinetic codes. Since these include the kinetic resonance this confirms the validity of the thermodynamic properties of our model. Furthermore, we can show that while a strongly nonlinear model is needed in kinetic theory a quasilinear model is sufficient in the fluid description. Thus our quasilinear fluid model will be adequate for treating all relevant problems in bulk transport. This is finally confirmed by the reproduction by the model of the experimental power scaling of the confinement time τE ∼ P−2/3. This confirms the validity of our reactive fluid model. This also gives credibility to our ITER simulations including the H-mode barrier. A new result is here, that alpha heating strongly reduces the slope of the H-mode barrier. This should significantly reduce the effects of ELM’s.