Gyrokinetic Turbulence Research Articles

Analysis of the Hermite spectrum in plasma turbulence

The properties of the Hermite spectrum associated with the linear drift-kinetic equation—as used in studies of gyrokinetic turbulence—are examined. A rigorous uniform asymptotic expression is derived for the steady-state spectrum with a Lenard-Bernstein collision operator. It is found that the spectrum is partitioned into three regions whose boundaries are determined by the ratio of the collision frequency ν to the parallel transit frequency kvth. In the regime of small Hermite index, n, with n ≲ (ν/kvth)2/3, collisions play no role, and the free energy decays like n−1/2 due to phase mixing. In the previously unexplored large-n regime, n≥(ν/kvth)2, collisions are dominant, and the decay of the free energy spectrum is extremely steep, falling off like (n/e)−n. Most of the free energy is dissipated in the intermediate regime, (ν/kvth)2/3 ≲ n≪(ν/kvth)2, where the asymptotic spectrum is in close agreement with the exponentially decaying “continuization” estimate. Our analysis shows that collisions act as a singular perturbation, giving rise to the intermediate regime, where collisions are significantly altering the spectrum well inside the general large-n asymptotic region.

Density gradient driven microinstabilities and turbulence in ASDEX Upgrade pellet fuelled plasmas

ASDEX Upgrade plasmas fuelled by pellets in the H-mode confinement regime are analyzed. The gyrokinetic code GKW is applied to calculate the microinstabilities which are predicted to be unstable in these plasmas. Two types of density gradient driven modes are found, outside and inside the pellet deposition location. The first mode is driven by a negative radial density gradient, and corresponds to the usual density gradient driven trapped electron mode instability, producing a large diffusive particle flux directed outwards, and becomes more unstable with increasing trapped particle fraction and with decreasing collisionality. The second is driven by a positive radial density gradient (that is, a locally hollow density profile) and is identified for the first time in this work. The instability is located in the proximity of the high field side of the poloidal cross section, and drives a diffusive particle flux directed inward. It is mainly produced by the non-adiabatic response of passing particles with low parallel velocities at high collisionality and it becomes more unstable with increasing passing particle fraction and increasing collision frequency. Nonlinear gyrokinetic turbulence simulations show that these instabilities can lead to saturated turbulence and produce particle diffusion at experimentally relevant levels. In contrast to the usual behavior of the turbulent fields in tokamak plasmas, which have largest fluctuations on the low field side, locally hollow density profiles are prediced to lead to turbulent electrostatic potential and density fluctuations which are maximum on the high field side of the torus.

Nonlinear gyrokinetic simulation of ion temperature gradient turbulence based on a numerical Lie-transform perturbation method

The electrostatic gyrokinetic nonlinear turbulence code NLT, which is based on a numerical Lie-transform perturbation method, is developed. For improving the computational efficiency and avoiding the numerical instabilities, field-aligned coordinates and a Fourier filter are adopted in the NLT code. Nonlinear tests of the ion temperature gradient driven turbulence with adiabatic electrons are performed for verifying the NLT code by comparing with other gyrokinetic codes. The time evolution of the ion heat diffusivity and the relation between the ion heat diffusivity and the ion temperature gradient are compared in the nonlinear tests. Good agreements are achieved from the nonlinear benchmarks between the NLT code and other codes. The mode structures of the perturbed electric potential representing different phases have been simulated.

Low frequency fully kinetic simulation of the toroidal ion temperature gradient instability

A fully kinetic ion model is useful for the verification of gyrokinetic turbulence simulations in certain regimes, where the gyrokinetic model may break down due to the lack of small ordering parameters. However, for a fully kinetic model to be of value, it must first be able to accurately simulate low frequency drift-type instabilities typically well within the domain of gyrokinetics. Here, a fully kinetic ion model is formulated with weak gradient drive terms and applied to the toroidal ion-temperature-gradient (ITG) instability for the first time. Implementation in toroidal geometry is discussed, where orthogonal coordinates are used for particle dynamics, but field-line-following coordinates are used for the field equation allowing for high resolution of the field-aligned mode structure. Variational methods are formulated for integrating the equation of motion allowing for accuracy at a modest time-step size. Linear results are reported for both the slab and toroidal ITG instabilities. Good agreement with full Vlasov and gyrokinetic theory is demonstrated in slab geometry. Good agreement with global gyrokinetic simulation is also shown in toroidal geometry.

Diagnosing collisionless energy transfer using field–particle correlations: gyrokinetic turbulence

Determining the physical mechanisms that extract energy from turbulent fluctuations in weakly collisional magnetized plasmas is necessary for a more complete characterization of the behaviour of a variety of space and astrophysical plasmas. Such a determination is complicated by the complex nature of the turbulence as well as observational constraints, chiefly that in situ measurements of such plasmas are typically only available at a single point in space. Recent work has shown that correlations between electric fields and particle velocity distributions constructed from single-point measurements produce a velocity-dependent signature of the collisionless damping mechanism. We extend this work by constructing field–particle correlations using data sets drawn from single points in strongly driven, turbulent, electromagnetic gyrokinetic simulations to demonstrate that this technique can identify the collisionless mechanisms operating in such systems. The velocity-space structure of the correlation between proton distributions and parallel electric fields agrees with expectations of resonant mechanisms transferring energy collisionlessly in turbulent systems. This work motivates the eventual application of field–particle correlations to spacecraft measurements in the solar wind, with the ultimate goal to determine the physical mechanisms that dissipate magnetized plasma turbulence.

A model of the saturation of coupled electron and ion scale gyrokinetic turbulence

A new paradigm of zonal flow mixing as the mechanism by which zonal E B fluctuations impact the saturation of gyrokinetic turbulence has recently been deduced from the nonlinear 2D spectrum of electric potential fluctuations in gyrokinetic simulations. These state of the art simulations span the physical scales of both ion and electron turbulence. It was found that the zonal flow mixing rate, rather than zonal flow shearing rate, competes with linear growth at both electron and ion scales. A model for saturation of the turbulence by the zonal flow mixing was developed and applied to the quasilinear trapped gyro-Landau fluid transport model (TGLF). The first validation tests of the new saturation model are reported in this paper with data from L-mode and high-βp regime discharges from the DIII-D tokamak. The shortfall in the predicted L-mode edge electron energy transport is improved with the new saturation model for these discharges but additional multiscale simulations are required in order to verify the safety factor and collisionality dependencies found in the modeling.

The development of magnetic field line wander in gyrokinetic plasma turbulence: dependence on amplitude of turbulence

The dynamics of a turbulent plasma not only manifests the transport of energy from large to small scales, but also can lead to a tangling of the magnetic field that threads through the plasma. The resulting magnetic field line wander can have a large impact on a number of other important processes, such as the propagation of energetic particles through the turbulent plasma. Here we explore the saturation of the turbulent cascade, the development of stochasticity due to turbulent tangling of the magnetic field lines and the separation of field lines through the turbulent dynamics using nonlinear gyrokinetic simulations of weakly collisional plasma turbulence, relevant to many turbulent space and astrophysical plasma environments. We determine the characteristic time $t_{2}$ for the saturation of the turbulent perpendicular magnetic energy spectrum. We find that the turbulent magnetic field becomes completely stochastic at time $t\lesssim t_{2}$ for strong turbulence, and at $t\gtrsim t_{2}$ for weak turbulence. However, when the nonlinearity parameter of the turbulence, a dimensionless measure of the amplitude of the turbulence, reaches a threshold value (within the regime of weak turbulence) the magnetic field stochasticity does not fully develop, at least within the evolution time interval $t_{2}<t\leqslant 13t_{2}$. Finally, we quantify the mean square displacement of magnetic field lines in the turbulent magnetic field with a functional form $\langle (\unicode[STIX]{x1D6FF}r)^{2}\rangle =A(z/L_{\Vert })^{p}$ ($L_{\Vert }$ is the correlation length parallel to the magnetic background field $\boldsymbol{B}_{\mathbf{0}}$, $z$ is the distance along $\boldsymbol{B}_{\mathbf{0}}$ direction), providing functional forms of the amplitude coefficient $A$ and power-law exponent $p$ as a function of the nonlinearity parameter.

Gyrokinetic turbulence: between idealized estimates and a detailed analysis of nonlinear energy transfers

Using large resolution numerical simulations of gyrokinetic (GK) turbulence, spanning an interval ranging from the end of the fluid scales to the electron gyroradius, we study the energy transfers in the perpendicular direction for a proton–electron plasma in a slab equilibrium magnetic geometry. The plasma parameters employed here are relevant to kinetic Alfvén wave turbulence in solar wind conditions. In addition, we use an idealized test representation for the energy transfers between two scales, to aid our understanding of the diagnostics applicable to the nonlinear cascade in an infinite inertial range. For GK turbulence, a detailed analysis of nonlinear energy transfers that account for the separation of energy exchanging scales is performed. Starting from the study of the energy cascade and the scale locality problem, we show that the general nonlocal nature of GK turbulence, captured via locality functions, contains a subset of interactions that are deemed local, are scale invariant (i.e. a sign of asymptotic locality) and possess a locality exponent that can be recovered directly from measurements on the energy cascade. It is the first time that GK turbulence is shown to possess an asymptotic local component, even if the overall locality of interactions is nonlocal. The results presented here and their implications are discussed from the perspective of previous findings reported in the literature and the idea of universality of GK turbulence.

Global anomalous transport of ICRH- and NBI-heated fast ions

By taking advantage of the trace approximation, one can gain an enormous computational advantage when solving for the global turbulent transport of impurities. In particular, this makes feasible the study of non-Maxwellian transport coupled in radius and energy, allowing collisions and transport to be accounted for on similar time scales, as occurs for fast ions. In this work, we study the fully-nonlinear ITG-driven trace turbulent transport of locally heated and injected fast ions. Previous results indicated the existence of MeV-range minorities heated by cyclotron resonance, and an associated density pinch effect. Here, we build upon this result using the t3core code to solve for the distribution of these minorities, consistently including the effects of collisions, gyrokinetic turbulence, and heating. Using the same tool to study the transport of injected fast ions, we contrast the qualitative features of their transport with that of the heated minorities. Our results indicate that heated minorities are more strongly affected by microturbulence than injected fast ions. The physical interpretation of this difference provides a possible explanation for the observed synergy when neutral beam injection (NBI) heating is combined with ion cyclotron resonance heating (ICRH). Furthermore, we move beyond the trace approximation to develop a model which allows one to easily account for the reduction of anomalous transport due to the presence of fast ions in electrostatic turbulence.

On the effect of neoclassical flows on intrinsic momentum in ASDEX Upgrade Ohmic L-mode plasmas

A gyro-kinetic analysis of intrinsic rotation is presented for the ASDEX Upgrade tokamak. The gyro-kinetic turbulence code, GKW and the neoclassical transport code, NEO are coupled so that the neoclassical equilibrium distribution function is included in the background distribution function in the gyro-kinetic turbulence simulation. This implementation is benchmarked against a similar implementation in the gyro-kinetic code, GS2 (Dorland et al 2000 Phys. Rev. Lett. 85 5579) and against analytical predictions.A quasi-linear and non-linear gyro-kinetic turbulence analysis is performed on Ohmic L-mode ASDEX Upgrade plasmas showing that the symmetry breaking effects due to neoclassical background flows can produce significant toroidal momentum transport. While its magnitude is of the order of other symmetry breaking mechanisms, such as the Coriolis pinch, up–down asymmetry in the magnetic flux surfaces and flow shear, the flow gradients it can sustain are appreciably smaller than the maximum gradients measured at the mid-radius of the ASDEX Upgrade tokamak core, which can be up to an order of magnitude larger.It is found that the gradient of the diamagnetic flow, and therefore the second derivatives of the density and temperature gradients are critical to the production of residual toroidal momentum flux. A quasi-linear estimate indicated that the second derivatives required to match the experimental flow gradient are up to an order of magnitude higher than the measured second derivatives. This analysis suggests that turbulent transport driven by neoclassical flows is not sufficient to explain the maximum flow gradients observed in ASDEX Upgrade.

What happens to full-f gyrokinetic transport and turbulence in a toroidal wedge simulation?

In order to save the computing time or to fit the simulation size into a limited computing hardware in a gyrokinetic turbulence simulation of a tokamak plasma, a toroidal wedge simulation may be utilized in which only a partial toroidal section is modeled with a periodic boundary condition in the toroidal direction. The most severe restriction in the wedge simulation is expected to be in the longest wavelength turbulence, i.e., ion temperature gradient (ITG) driven turbulence. The global full-f gyrokinetic code XGC1 is used to compare the transport and turbulence properties from a toroidal wedge simulation against the full torus simulation in an ITG unstable plasma in a model toroidal geometry. It is found that (1) the convergence study in the wedge number needs to be conducted all the way down to the full torus in order to avoid a false convergence, (2) a reasonably accurate simulation can be performed if the correct wedge number N can be identified, (3) the validity of a wedge simulation may be checked by performing a wave-number spectral analysis of the turbulence amplitude |δΦ| and assuring that the variation of δΦ between the discrete kθ values is less than 25% compared to the peak |δΦ|, and (4) a frequency spectrum may not be used for the validity check of a wedge simulation.

Benchmarking the GENE and GYRO codes through the relative roles of electromagnetic and E × B stabilization in JET high-performance discharges

Nonlinear gyrokinetic simulations using the GENE code have previously predicted a significant nonlinear enhanced electromagnetic stabilization in certain JET discharges with high neutral-beam power and low core magnetic shear (Citrin et al 2013 Phys. Rev. Lett. 111 155001, 2015 Plasma Phys. Control. Fusion 57 014032). This dominates over the impact of E × B flow shear in these discharges. Furthermore, fast ions were shown to be a major contributor to the electromagnetic stabilization. These conclusions were based on results from the GENE gyrokinetic turbulence code. In this work we verify these results using the GYRO code. Comparing results (linear frequencies, eigenfunctions, and nonlinear fluxes) from different gyrokinetic codes as a means of verification (benchmarking) is only convincing if the codes agree for more than one discharge. Otherwise, agreement may simply be fortuitous. Therefore, we analyze three discharges, all with a carbon wall: a simplified, two-species, circular geometry case based on an actual JET discharge; an L-mode discharge with a significant fast-ion pressure fraction; and a low-triangularity high-β hybrid discharge. All discharges were analyzed at normalized toroidal flux coordinate ρ = 0.33 where significant ion temperature peaking is observed. The GYRO simulations support the conclusion that electromagnetic stabilization is strong, and dominates E × B shear stabilization.

Publisher's Note: “The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence” [Phys. Plasmas 23, 062518 (2016)

Publisher's Note: “The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence” [Phys. Plasmas <b>23</b>, 062518 (2016)

Microtearing turbulence limiting the JET-ILW pedestal

The first nonlinear gyrokinetic turbulence simulations that quantitatively reproduce experimental transport levels in an H-mode pedestal are reported. In the JET-ILW (ITER-like wall) pedestal, the bulk of the transport in the steep gradient region is caused by the turbulence driven by the microtearing mode (MTM). Kinetic ballooning modes are found to be in a second-stability regime. With contributions from the neoclassical and electron temperature gradient driven transport, the MTM mechanism reproduces, quantitatively, the experimental power balance across most of the pedestal.

Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra

A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradient-driven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest in the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeit in ways that cannot be fully explained by the linear eigenvalue approach; a non-modal treatment is necessary to understand key features of the turbulence.

The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence

The 2D spectrum of the saturated electric potential from gyrokinetic turbulence simulations that include both ion and electron scales (multi-scale) in axisymmetric tokamak geometry is analyzed. The paradigm that the turbulence is saturated when the zonal (axisymmetic) ExB flow shearing rate competes with linear growth is shown to not apply to the electron scale turbulence. Instead, it is the mixing rate by the zonal ExB velocity spectrum with the turbulent distribution function that competes with linear growth. A model of this mechanism is shown to be able to capture the suppression of electron-scale turbulence by ion-scale turbulence and the threshold for the increase in electron scale turbulence when the ion-scale turbulence is reduced. The model computes the strength of the zonal flow velocity and the saturated potential spectrum from the linear growth rate spectrum. The model for the saturated electric potential spectrum is applied to a quasilinear transport model and shown to accurately reproduce the electron and ion energy fluxes of the non-linear gyrokinetic multi-scale simulations. The zonal flow mixing saturation model is also shown to reproduce the non-linear upshift in the critical temperature gradient caused by zonal flows in ion-scale gyrokinetic simulations.

Comparison of gradient and flux driven gyro-kinetic turbulent transport

Flux and gradient driven ion temperature gradient turbulence in tokamak geometry and for Cyclone base case parameters are compared in the local limit using the same underlying gyro-kinetic turbulence model. The gradient driven turbulence described using the flux tube model with periodic boundary conditions has a finite ion heat flux Qi≈10n0T0ρ*2vth, where n0 (T0) is the background density (temperature), ρ*=ρ/R is the normalized Larmor radius, R is the major radius of the device, and vth is the ion thermal velocity at the nonlinear threshold of the temperature gradient length for turbulence generation. Consequently, the gradient driven local transport model is unable to accurately describe heat fluxes below Qi&amp;lt;10n0T0ρ*2vth, since no stationary fully developed turbulent state can be obtained. The turbulence in the flux driven case shows intermittent behaviour and avalanches for Qi&amp;lt;10n0T0ρ*2vth. Isolated avalanches disappear for Qi&amp;gt;10n0T0ρ*2vth, and at higher heat fluxes, the statistics of the turbulence is the same for the flux and gradient driven case. The nonlinear upshift of the temperature gradient length threshold for turbulence generation (known as the Dimits shift) is larger in the case of flux driven turbulence. This higher nonlinear upshift is attributed to the generation of structures in the radial temperature profile, known as staircases [Dif-Pradalier, Phys. Rev. E 82, 025401 (2010)]. Avalanches are initiated at specific locations and have roughly the same radial extent of 50–70 ion Larmor radii. The staircases are obtained at low heating rates, and become unstable and break up at higher heating rates. At the heat fluxes for which staircase formation is observed, no stationary gradient driven simulations can be obtained.

The anisotropic redistribution of free energy for gyrokinetic plasma turbulence in a Z-pinch

For a Z-pinch geometry, we report on the nonlinear redistribution of free energy across scales perpendicular to the magnetic guide field, for a turbulent plasma described in the framework of gyrokinetics. The analysis is performed using a local flux-surface approximation, in a regime dominated by electrostatic fluctuations driven by the entropy mode, with both ion and electron species being treated kinetically. To explore the anisotropic nature of the free energy redistribution caused by the emergence of zonal flows, we use a polar coordinate representation for the field-perpendicular directions and define an angular density for the scale flux. Positive values for the classically defined (angle integrated) scale flux, which denote a direct energy cascade, are shown to be also composed of negative angular sections, a fact that impacts our understanding of the backscatter of energy and the way in which it enters the modeling of sub-grid scales for turbulence. A definition for the flux of free energy across each perpendicular direction is introduced as well, which shows that the redistribution of energy in the presence of zonal flows is highly anisotropic.

Mesh generation for confined fusion plasma simulation

XGC1 and M3D-C1 are two fusion plasma simulation codes being developed at Princeton Plasma Physics Laboratory. XGC1 uses the particle-in-cell method to simulate gyrokinetic neoclassical physics and turbulence (Chang et al. Phys Plasmas 16(5):056108, 2009; Ku et al. Nucl Fusion 49:115021, 2009; Admas et al. J Phys 180(1):012036, 2009). M3D-$$C^1$$C1 solves the two-fluid resistive magnetohydrodynamic equations with the $$C^1$$C1 finite elements (Jardin J comput phys 200(1):133---152, 2004; Jardin et al. J comput Phys 226(2):2146---2174, 2007; Ferraro and Jardin J comput Phys 228(20):7742---7770, 2009; Jardin J comput Phys 231(3):832---838, 2012; Jardin et al. Comput Sci Discov 5(1):014002, 2012; Ferraro et al. Sci Discov Adv Comput, 2012; Ferraro et al. International sherwood fusion theory conference, 2014). This paper presents the software tools and libraries that were combined to form the geometry and automatic meshing procedures for these codes. Specific consideration has been given to satisfy the mesh configuration and element shape quality constraints of XGC1 and M3D-$$C^1$$C1.

Cross-Scale Interactions between Electron and Ion Scale Turbulence in a Tokamak Plasma.

Multiscale gyrokinetic turbulence simulations with the real ion-to-electron mass ratio and β value are realized for the first time, where the β value is given by the ratio of plasma pressure to magnetic pressure and characterizes electromagnetic effects on microinstabilities. Numerical analysis at both the electron scale and the ion scale is used to reveal the mechanism of their cross-scale interactions. Even with the real-mass scale separation, ion-scale turbulence eliminates electron-scale streamers and dominates heat transport, not only of ions but also of electrons. Suppression of electron-scale turbulence by ion-scale eddies, rather than by long-wavelength zonal flows, is also demonstrated by means of direct measurement of nonlinear mode-to-mode coupling. When the ion-scale modes are stabilized by finite-β effects, the contribution of the electron-scale dynamics to the turbulent transport becomes non-negligible and turns out to enhance ion-scale turbulent transport. Damping of the ion-scale zonal flows by electron-scale turbulence is responsible for the enhancement of ion-scale transport.