Electrostatic Drift Wave Turbulence Research Articles

Scientific machine learning based reduced-order models for plasma turbulence simulations

This paper investigates non-intrusive Scientific Machine Learning (SciML) Reduced-Order Models (ROMs) for plasma turbulence simulations. In particular, we focus on Operator Inference (OpInf) to build low-cost physics-based ROMs from data for such simulations. As a representative example, we consider the (classical) Hasegawa–Wakatani (HW) equations used for modeling two-dimensional electrostatic drift-wave turbulence. For a comprehensive perspective of the potential of OpInf to construct predictive ROMs, we consider three setups for the HW equations by varying a key parameter, namely, the adiabaticity coefficient. These setups lead to the formation of complex and nonlinear dynamics, which makes the construction of predictive ROMs of any kind challenging. We generate the training datasets by performing direct numerical simulations of the HW equations and recording the computed state data and outputs over a time horizon of 100 time units in the turbulent phase. We then use these datasets to construct OpInf ROMs for predictions over 400 additional time units, that is, 400% more than the training horizon. Our results show that the OpInf ROMs capture important statistical features of the turbulent dynamics and generalize beyond the training time horizon while reducing the computational effort of the high-fidelity simulation by up to five orders of magnitude. In the broader context of fusion research, this shows that non-intrusive SciML ROMs have the potential to drastically accelerate numerical studies, which can ultimately enable tasks such as the design of optimized fusion devices.

Theory of the Tertiary Instability and the Dimits Shift from Reduced Drift-Wave Models.

Tertiary modes in electrostatic drift-wave turbulence are localized near extrema of the zonal velocity U(x) with respect to the radial coordinate x. We argue that these modes can be described as quantum harmonic oscillators with complex frequencies, so their spectrum can be readily calculated. The corresponding growth rate γ_{TI} is derived within the modified Hasegawa-Wakatani model. We show that γ_{TI} equals the primary-instability growth rate plus a term that depends on the local U^{''}; hence, the instability threshold is shifted compared to that in homogeneous turbulence. This provides a generic explanation of the well-known yet elusive Dimits shift, which we find explicitly in the Terry-Horton limit. Linearly unstable tertiary modes either saturate due to the evolution of the zonal density or generate radially propagating structures when the shear |U^{'}| is sufficiently weakened by viscosity. The Dimits regime ends when such structures are generated continuously.

A mechanism of neoclassical tearing modes onset by drift wave turbulence

The evolution of neoclassical tearing modes (NTMs) in the presence of electrostatic drift wave turbulence is investigated. In contrast with anomalous transport effect induced by turbulence on NTMs, a new mechanism that turbulence-driven current can affect the onset threshold of NTMs significantly is suggested. Turbulence acts as a source or sink to exchange energy with NTMs. The turbulence-driven current can change the parallel current in magnetic islands and affect the evolution of NTMs, depending on the direction of turbulence intensity gradient. When the turbulence intensity gradient is negative, the turbulence-driven current enhances the onset threshold of NTMs. When the turbulence intensity gradient is positive, it can reduce or even overcome the stabilizing effect of neoclassical polarization current, leading to a small onset threshold of NTMs. This implies that NTMs can appear without noticeable magnetohydrodynamics (MHD) events.

Generation of electrostatic drift zonal flows under the action of mean sheared flows

Generation of large-scale zonal flows by the small-scale electrostatic drift wave turbulence in the magnetized plasma under the action of mean poloidal sheared flow is considered. Attention to large-scale (compared to the ion Larmor radius) drift structures is paid. To this end, the generalized Hasegawa-Mima equation containing both vector and scalar nonlinearities is derived, and the appropriate eigenvalue problem is solved numerically. Destabilizing role of the small amplitude mean shear flow and spatial inhomogeneity of electron temperature is shown.

Spectral properties of electrostatic drift wave turbulence in the laboratory and the ionosphere

Abstract. Low-frequency electrostatic drift wave turbulence has been studied in both laboratory plasmas and in space. The present review describes a number of such laboratory experiments together with results obtained by instrumented spacecraft in the Earth's near and distant ionospheres. The summary emphasizes readily measurable quantities, such as the turbulent power spectra for the fluctuations in plasma density, potential and electric fields. The agreement between power spectra measured in the laboratory and in space seems to be acceptable, but there are sufficiently frequent counterexamples to justify a future dedicated analysis, for instance by numerical tools, to explain deviations. When interpreting spectra at low ionospheric altitudes, it is necessary to give attention to the DC ionospheric electric fields and the differences in the physics of electron–ion collisions and collisions of charged particles with neutrals for cases with significant Hall drifts. These effects modify the drift wave spectra. A dedicated laboratory experiment accounted for some of these differences.

Lattice Boltzmann model for collisionless electrostatic drift wave turbulence obeying Charney–Hasegawa–Mima dynamics

A lattice Boltzmann method (LBM) approach to the Charney–Hasegawa–Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occurring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.

The temporal evolution of the kinetic drift-Alfven instability of plasma shear flow

The linear non-modal kinetic theory of the kinetic drift-Alfven instability, in the presence of shearing plasma flow, exhibits temporal non-modal amplitude growth with time-increasing growth rate. The drift-Alfven instability develops when the evolving frequencies of the density-gradient-driven drift wave and the kinetic Alfven wave become comparable and mode coupling increases significantly. Because the development of the instability depends on the plasma density gradient, drift-Alfven turbulence may be invoked as a possible turbulence driver in the edge pedestal, limiting the pedestal density gradients during the edge localized modes crash. In contrast to the case of shear-modified electrostatic drift-wave turbulence, the scattering of plasma ions by the shear-modified electromagnetic drift-Alven turbulence does not only suppress the turbulence, but can be responsible for transient increase in the instability's growth rate. The instability, i.e., turbulence, stabilizes when the drift and Alfven wave frequencies eventually become different enough that effective coupling between the two modes ends. So, perpendicular-flow shear can cause the growth rate to increase, decrease, become zero, and become negative as a result of shear's time-changing modification to the frequencies of the drift and Alfven waves.

Properties of microturbulence in toroidal plasmas with reversed magnetic shear

Electrostatic drift wave turbulence in tokamak plasmas with reversed magnetic shear is studied using global gyrokinetic particle simulations. The linear eigenmode of the ion temperature gradient (ITG) instability exhibits a mode gap around the minimum safety factor (qmin) region, particularly when qmin is an integer, due to the rarefaction of rational surfaces. The collisionless trapped electron mode (CTEM) instability is suppressed in the negative-shear region due to the reversal of the toroidal precessional drift of trapped electrons. However, after nonlinear saturation, the ITG gap is filled up by the turbulence spreading and the CTEM fluctuation propagates into the stable negative-shear region. The steady state turbulence occupies the whole volume without any identifiable gap or coherent structures of the heat conductivity, perturbed temperature, or zonal flows in the qmin location or the reversed shear region. Our finding indicates that the electrostatic drift wave turbulence itself does not support either linear or nonlinear mechanism for the formation of internal transport barriers in the reversed magnetic shear when qmin crossing an integer.

Multitude of Core-Localized Shear Alfvén Waves in a High-Temperature Fusion Plasma

Evidence is presented for a multitude of discrete frequency Alfvén waves in the core of magnetically confined high-temperature fusion plasmas. Multiple diagnostic instruments confirm wave excitation over a wide spatial range from the device size at the longest wavelengths down to the thermal ion Larmor radius. At the shortest scales, the poloidal wavelengths are comparable to the scale length of electrostatic drift wave turbulence. Theoretical analysis confirms a dominant interaction of the modes with particles in the thermal ion distribution traveling well below the Alfvén velocity.

Turbulent flux and the diffusion of passive tracers in electrostatic turbulence

The connection between the diffusion of passive tracer particles and the anomalous turbulent flux in electrostatic drift-wave turbulence is investigated by direct numerical solutions of the 2D Hasegawa–Wakatani equations. The probability density functions for the point-wise and flux surface averaged turbulent particle flux are measured and compare well to a folded Gaussian, respectively a log-normal distribution. By following a large number of passive tracer particles we evaluate the diffusion coefficient based on the particle dispersion. It is found that the particle diffusion coefficient is in good agreement with the one derived from the turbulent E×B-flux by using Fick’s law. Employing the Lagrangian conservation of the “Potential Vorticity” in the Hasegawa–Wakatani equations, the analytical support for this result is obtained. The transport estimated by passive tracer dispersion and turbulent plasma flux are found to coincide.

Particle diffusion in anisotropic turbulence

We have investigated the diffusion of test particles in anisotropic two-dimensional turbulence. To be specific we have considered electrostatic drift-wave turbulence in magnetized plasma, described by the so-called Hasegawa–Wakatani equations, which produce a two-dimensional turbulent flow-field with a self-consistent instability drive. However, our results are also relevant for particle dispersion in beta-plane turbulence, due to the similarity between drift waves in plasmas and Rossby waves on the beta-plane. In general, we observe that the particle dispersion is strongly anisotropic, with the diffusion in the zonal direction much larger than the diffusion in the direction parallel to the background density gradient––the radial direction (meridional direction for the beta-plane analogy). Trapping of particles in and subsequent displacement with vortical structures leads to an intermediate anomalous diffusion with superdiffusive behaviour in the zonal direction and subdiffusive behaviour in the radial direction. However, the time asymptotic behaviour always approaches a normal diffusive process. By including inertial effects in the particle velocity we observe a tendency for increased superdiffusive behaviour.

Drift Wave Based Anomalous Transport Models

Energy and particle transport rates in magnetically confined plasmas are often larger than neo-classical transport owing to binary collisions would allow. Anomalous transport, a major road block on the path to an economic fusion reactor, is a consequence of electric and magnetic fluctuations driven to supra thermal levels by various instability mechanisms. The linearly excited modes saturate by inducing a relaxation of the equilibrium profiles towards the marginally stable state, on the one hand, and via various non-linear interaction mechanisms, on the other hand. Specific instabilities, profile relaxation and non-linear interaction models are described and their successes and drawbacks are analysed in the light of observed characteristics of plasma confinement. A rough evaluation of the nuclear heating power required to balance the anomalous losses in the International Tokamak Experimental Reactor (ITER) is derived on the basis of the very qualitative mixing length estimate applied to electrostatic drift wave turbulence. Results from large-scale gyro-kinetic simulation codes are discussed.

In search of zonal flows using cross-bispectrum analysis in the boundary plasma of the Hefei Tokamak-7

Langmuir probes have been used to measure the electrostatic Reynolds stress and the floating potential fluctuation in the boundary plasma of the Hefei Tokamak-7 (HT-7) [J. Li, B. N. Wan, and J. S. Mao, Plasma Phys. Controlled Fusion 42, 135 (2000)]. The cross bispectrum of 〈ṼrṼθφ̃f〉 indicates the existence of difference-frequency nonlinear phase coupling and the generation of fluctuations near the geodesic acoustic mode frequency. The inverse cascade process might be linked to the generation of zonal flows by small-scale electrostatic drift-wave turbulence.

A Multigrid Algorithm for Nonlocal Collisional Electrostatic Drift-Wave Turbulence

We have developed a three-dimensional anisotropic multigrid solver for simulating nonlocal collisional electrostatic drift-wave turbulence in a tokamak with magnetic shear. As an example, the solver has been used to obtain entire flux-surface solutions of the nonlocal Hasegawa–Wakatani equations in the absence of curvature effects. The implicit treatment of the parallel-gradient terms permits the use of a relatively large time step. Considerable effort was made in the design of the implicit solver to ensure that the presence of anisotropy does not lead to a significant degradation in performance. The multigrid algorithm has several advantages over a pseudospectral Poisson solver; most importantly, all nonlinear terms, including those in the Ohm's law, can be retained in a straightforward manner. Although in this work the solver is illustrated using straightened tokamak (sheared slab) geometry, the object-oriented construction of the code will facilitate the eventual inclusion of curvature terms and the complete nonlinear reduced Braginskii equations, including ion thermal dynamics.

Dispersion of ideal particles in a two-dimensional model of electrostatic turbulence

The dispersion of ideal test particles in electrostatic drift-wave turbulence is investigated numerically. A self-consistent model with an internal instability drive is used to obtain the turbulent two-dimensional (2D) flow-field. It is shown that nonlinear couplings lead to the formation of coherent vortical structures in the flow. The dispersion of the particles is found to be anisotropic, with the weakest dispersion in the direction of the density gradient. By distinguishing between particles trapped in structures and free particles, it is demonstrated that the trapping and subsequent displacement of particles by nonlinear vortex structures enhances the particle diffusion in the direction of the background density gradient. Conditional diffusion coefficients are obtained showing that particles trapped by the vortex structures are convected by the structures. The time a particle on the average stays trapped in the structure is closely related to the lifetime of the vortical structures. The relation between the diffusion coefficient obtained from the test particle dispersion and an effective diffusion coefficient obtained from the cross-field turbulent flux is discussed.

The effect of sheared diamagnetic flow on turbulent structures generated by the Charney–Hasegawa–Mima equation

The generation of electrostatic drift wave turbulence is modeled by the Charney–Hasegawa–Mima equation. The equilibrium density gradient n0=n0(x) is chosen so that dn0/dx is nonzero and spatially variable (i.e., v*e is sheared). It is shown that this sheared diamagnetic flow leads to localized turbulence which is concentrated at max(∇n0), with a large dv*e/dx inhibiting the spread of the turbulence in the x direction. Coherent structures form which propagate with the local v*e in the y direction. Movement in the x direction is accompanied by a change in their amplitudes. When the numerical code is initialized with a single wave, the plasma behavior is dominated by the initial mode and its harmonics.

Density behaviour and particle losses in RF discharge plasmas of the URAGAN-3M torsatron

URAGAN-3M (U-3M) is an l=3, m=9 torsatron with R0=1 m, ā ≈ 12 cm and ϵ(ā) ≈ 0.4. The entire magnetic system is enclosed in a large vacuum chamber and there is no material limiter, so that an open helical divertor is realized in this device. A plasma is RF produced in 100% hydrogen gas and heated in the multimode Alfven resonance regime (ω ≲ ωci) at a continuous working gas leak-in. With an RF power of ~200 kW at B = 0.45 T, a quasi-steady (up to 50 ms) state of the plasma parameters sets in with n̄e ≃ 2 × 1018 m-3, Ti(0) ≃ 130 eV and Te(0) equivalent to 300 eV. By using microwaves and Langmuir probes, the time and space behaviours of the electron density/ion saturation current are studied in the confinement volume, near the boundary of the confinement region, and in the divertor flux region. On the basis of ion saturation current profile measurements near the boundary of the confinement region the values of local particle flux density across the boundary are estimated: Γ⊥ ≈ 7 × 1020 m-2.s-1 far from the antenna and Γ⊥ approximately=1021 m-2.s-1, Γ⊥ ≈ 2 × 1021 m-2.s-1 near the antenna. Such flux densities and the corresponding diffusion coefficients are anomalously high, exceeding the neoclassical values by 2 to 3 orders of magnitude. On the other hand, these values are comparable with the flux densities and diffusivities obtained in edge plasma with similar parameters on other stellarators under NBI and ECR heating conditions. Therefore, it is claimed that the method of plasma heating that has been chosen in U-3M does not result in any substantial deterioration of particle confinement (with the possible exception of the region close to the antenna) as against other methods. Presumably, the anomalous particle transport across the boundary in U-3M is caused by destruction of magnetic surfaces and/or electrostatic drift wave turbulence. It is found thatincreasing the RF power results in an increase of the diverted plasma flow and of the hydrogen consumptionnecessary to maintain the average plasma density at a given level. This is considered as a manifestation of acommon regularity, according to which particle losses grow with heating power, this effect having been observedrecently on a stellarator type device under NBI and ECR heating conditions. An effect specific to U-3M, theincrease of the average electron density, being observed within 1 to 2 ms after RF pulse termination, is attributedmainly to particle transport reduction due to plasma cooling, while the intensity of neutral particle ionizationstill remains sufficiently high in the confinement volume at the initial phase of cooling under continuous hydrogenleak-in conditions.

Three-dimensional modelling of drift-wave turbulence

An EPIC (Extended Particle-in-Cell) scheme is developed to model electrostatic drift-wave turbulence in three space dimensions. An existing two-dimensional code that treats magnetic shear and non-adiabatic effects is adapted to incorporate the ICEx algorithm, that enables instability-free numerical modelling of complex wave phenomena without a timestep restriction. Successful simulations of drift-wave interactions are presented.

Particle-Mesh Schemes for Advection Dominated Flows

Extended particle-in-cell (EPIC) schemes are considered with a view to applications in electrostatic drift-wave turbulence and ordinary hydrodynamical turbulence, where periodic boundary conditions are inappropriate. We treat issues relating to the dual particle-mesh representation and to the need to follow particle orbits accurately. A successful application of EPIC to an advection dominated flow is demonstrated. The errors are quantified so that choosing suitable numerical parameters to obtain a result of a given accuracy is straightforward.

Particle-mesh modelling of drift-wave turbulence

An EPIC (Extended Particle-in-Cell) scheme is developed to model electrostatic drift-wave turbulence in two space dimensions. The efficient inclusion of additional physical effects such as magnetic shear and non-adiabaticity is treated. The algorithm is set out at length together with a numerical stability analysis. Successful simulations of drift-wave interactions are presented: fluctuation level and particle transport are found to be strongly correlated. Finally, arguments favouring the use of EPIC for drift-wave studies are put forward.