Gyrokinetic Turbulence Simulations Research Articles

Bringing global gyrokinetic turbulence simulations to the transport timescale using a multiscale approach

The vast separation dividing the characteristic times of energy confinement and turbulence in the core of toroidal plasmas makes first-principles prediction on long timescales extremely challenging. Here we report the demonstration of a multiple-timescale method that enables coupling global gyrokinetic simulations with a transport solver to calculate the evolution of the self-consistent temperature profile. This method, which exhibits resiliency to the intrinsic fluctuations arising in turbulence simulations, holds potential for integrating nonlocal gyrokinetic turbulence simulations into predictive, whole-device models.

Validation of gyrokinetic simulations with measurements of electron temperature fluctuations and density-temperature phase angles on ASDEX Upgrade

Measurements of turbulent electron temperature fluctuation amplitudes, δTe⊥/Te, frequency spectra, and radial correlation lengths, Lr(Te⊥), have been performed at ASDEX Upgrade using a newly upgraded Correlation ECE diagnostic in the range of scales k⊥<1.4 cm−1, kr<3.5 cm−1 (k⊥ρs<0.28 and krρs<0.7). The phase angle between turbulent temperature and density fluctuations, αnT, has also been measured by using an ECE radiometer coupled to a reflectometer along the same line of sight. These quantities are used simultaneously to constrain a set of ion-scale non-linear gyrokinetic turbulence simulations of the outer core (ρtor = 0.75) of a low density, electron heated L-mode plasma, performed using the gyrokinetic simulation code, GENE. The ion and electron temperature gradients were scanned within uncertainties. It is found that gyrokinetic simulations are able to match simultaneously the electron and ion heat flux at this radius within the experimental uncertainties. The simulations were performed based on a reference discharge for which δTe⊥/Te measurements were available, and Lr(Te⊥) and αnT were then predicted using synthetic diagnostics prior to measurements in a repeat discharge. While temperature fluctuation amplitudes are overestimated by >50% for all simulations within the sensitivity scans performed, good quantitative agreement is found for Lr(Te⊥) and αnT. A validation metric is used to quantify the level of agreement of individual simulations with experimental measurements, and the best agreement is found close to the experimental gradient values.

Identifying microturbulence regimes in a TCV discharge making use of physical constraints on particle and heat fluxes

Reducing the uncertainty on physical input parameters derived from experimental measurements is essential towards improving the reliability of gyrokinetic turbulence simulations. This can be achieved by introducing physical constraints. Amongst them, the zero particle flux condition is considered here. A first attempt is also made to match as well the experimental ion/electron heat flux ratio. This procedure is applied to the analysis of a particular Tokamak à Configuration Variable discharge. A detailed reconstruction of the zero particle flux hyper-surface in the multi-dimensional physical parameter space at fixed time of the discharge is presented, including the effect of carbon as the main impurity. Both collisionless and collisional regimes are considered. Hyper-surface points within the experimental error bars are found. The analysis is done performing gyrokinetic simulations with the local version of the GENE code, computing the fluxes with a Quasi-Linear (QL) model and validating the QL results with non-linear simulations in a subset of cases.

Suppression of Ion-Scale Microtearing Modes by Electron-Scale Turbulence via Cross-Scale Nonlinear Interactions in Tokamak Plasmas

Gyrokinetic turbulence simulations are applied for the first time to the cross-scale interactions of microtearing modes (MTMs) and electron-temperature-gradient (ETG) modes. The investigation of the fluctuation response in a multiscale simulation including both types of instabilities indicates that MTMs are suppressed by ETG turbulence. A detailed analysis of nonlinear mode coupling reveals that radially localized current-sheet structures of MTMs are strongly distorted by fine-scale E×B flows of ETG turbulence. Consequently, electron heat transport caused by the magnetic flutter of MTMs is significantly reduced and ETG turbulence dominates electron heat transport.

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.

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.

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.

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.

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.

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.

Multispecies density peaking in gyrokinetic turbulence simulations of low collisionality Alcator C-Mod plasmas

Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.

Testing the high turbulence level breakdown of low-frequency gyrokinetics against high-frequency cyclokinetic simulationsa)

This paper presents numerical simulations of the nonlinear cyclokinetic equations in the cyclotron harmonic representation [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 (2013)]. Simulations are done with a local flux-tube geometry and with the parallel motion and variation suppressed using a newly developed rCYCLO code. Cyclokinetic simulations dynamically follow the high-frequency ion gyro-phase motion which is nonlinearly coupled into the low-frequency drift-waves possibly interrupting and suppressing gyro-averaging and increasing the transport over gyrokinetic levels. By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the breakdown of gyrokinetics at high turbulence levels is quantitatively tested over a range of relative ion cyclotron frequency 10 < Ω* < 100 where Ω* = 1/ρ*, and ρ* is the relative ion gyroradius. The gyrokinetic linear mode rates closely match the cyclokinetic low-frequency rates for Ω* > 5. Gyrokinetic transport recovers cyclokinetic transport at high relative ion cyclotron frequency (Ω* ≥ 50) and low turbulence level as required. Cyclokinetic transport is found to be lower than gyrokinetic transport at high turbulence levels and low-Ω* values with stable ion cyclotron (IC) modes. The gyrokinetic approximation is found to break down when the density perturbations exceed 20%. For cyclokinetic simulations with sufficiently unstable IC modes and sufficiently low Ω* ∼ 10, the high-frequency component of cyclokinetic transport level can exceed the gyrokinetic transport level. However, the low-frequency component of the cyclokinetic transport and turbulence level does not exceed that of gyrokinetics. At higher and more physically relevant Ω* ≥ 50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport and turbulence level is still smaller than that of gyrokinetics. Thus, the cyclokinetic simulations do not account for the so-called “L-mode near edge short fall” seen in some low-frequency gyrokinetic transport and turbulence simulations.

Effect of turbulence on electron cyclotron current drive and heating in ITER

Non-linear local electromagnetic gyrokinetic turbulence simulations of the ITER standard scenario H-mode are presented for the q = 3/2 and q = 2 surfaces. The turbulent transport is examined in regions of velocity space characteristic of electrons heated by electron-cyclotron waves. Electromagnetic fluctuations and sub-dominant micro-tearing modes are found to contribute significantly to the transport of the accelerated electrons, even though they have only a small impact on the transport of the bulk species. The particle diffusivity for resonant passing electrons is found to be less than 0.15 m2 s−1, and their heat conductivity is found to be less than 2 m2 s−1. Implications for the broadening of the current drive and energy deposition in ITER are discussed.

Resolving the mystery of transport within internal transport barriers

The Trapped Gyro-Landau Fluid (TGLF) quasi-linear model [G. M. Staebler, et al., Phys. Plasmas 12, 102508 (2005)], which is calibrated to nonlinear gyrokinetic turbulence simulations, is now able to predict the electron density, electron and ion temperatures, and ion toroidal rotation simultaneously for internal transport barrier (ITB) discharges. This is a strong validation of gyrokinetic theory of ITBs, requiring multiple instabilities responsible for transport in different channels at different scales. The mystery of transport inside the ITB is that momentum and particle transport is far above the predicted neoclassical levels in apparent contradiction with the expectation from the theory of suppression of turbulence by E×B velocity shear. The success of TGLF in predicting ITB transport is due to the inclusion of ion gyro-radius scale modes that become dominant at high E×B velocity shear and to improvements to TGLF that allow momentum transport from gyrokinetic turbulence to be faithfully modeled.

A new paradigm for E × B velocity shear suppression of gyro-kinetic turbulence and the momentum pinch

Detailed studies of the non-linear radial wavenumber spectrum of electric potential fluctuations in gyro-kinetic plasma turbulence simulations have led to a new paradigm that is capable of computing the momentum pinch quasilinearly. It is found that shear in the E × B velocity Doppler shift suppresses turbulence by inducing a shift in the peak of the radial wavenumber spectrum, and a reduction in the amplitude. An analytic model of the process is used to understand the roles of the sheared velocity and the non-linear mode coupling. The analytic model leads to a simple formula that fits the non-linear spectrum and only depends on the spectral average shift in the radial wavenumber. This ‘spectral shift’ model is a new paradigm that radial parity breaking is the fundamental mechanism that suppresses the turbulence through a radial wavenumber shift. The E × B velocity shear is one of a number of radial parity breaking mechanisms. Using a model of the spectral shift the toroidal Reynolds stress due to the E × B velocity shear can be computed for the first time with a quasilinear model. It is shown that, when diamagnetic and neoclassical contributions to the parallel flows are included, the E × B velocity shear term in the toroidal Reynolds stress allows the sign of the intrinsic toroidal rotation to change. Simulations of the co-current and balanced neutral beam injection phase of a DIII-D discharge using the quasilinear model show good agreement with experiment.

Improved Multi-Mode anomalous transport module for tokamak plasmas

The Multi-Mode anomalous transport module version 7.1 (MMM7.1) is a theory-based transport model that is used to predict temperature, density and rotation profiles for tokamak plasmas in integrated whole device modeling codes. The theoretical foundation of the current version, MMM7.1, has been significantly advanced since the first released version in 1995, MMM95. The latest version of the Multi-Mode model, MMM7.1, includes an improved Weiland model for the ITG, TEM, and MHD modes, the Horton model for short wavelength ETG modes and the Rafiq model for the drift resistive inertial ballooning modes (DRIBMs). The ETG transport threshold in the Horton model is refined by using the threshold obtained from toroidal gyrokinetic ETG turbulence simulations. The different components of the MMM7.1 model provide contributions to transport in the different regions of plasma discharge. To facilitate the implementation of the latest version of the Multi-Mode module in integrated predictive modeling codes, a clearly specified interface is described and a test program is provided in order to examine the predictions provided by MMM7.1. MMM7.1 is documented and organized as a standalone module, which fully complies with the National Transport Code Collaboration (NTCC) standards. The MMM7.1 module has been used both with a standalone driver program as well as within the PTRANSP code. Results are presented to illustrate the extent to which the various component models contribute to transport in both L-mode and H-mode discharges.

Gyrokinetic turbulence simulations of high-beta tokamak and helical plasmas with full-kinetic and hybrid models

Turbulent transport in high-beta toroidal plasmas is investigated by means of an electromagnetic gyrokinetic model and a newly developed electromagnetic hybrid model consisting of the gyrokinetic equation for ions and drift-Landau-fluid equations for electrons. Full gyrokinetic simulation results for Cyclone base case tokamak and for Large Helical Device (LHD) plasmas are quickly and accurately reproduced by the hybrid simulation. In the kinetic ballooning mode (KBM)-driven turbulence the ion heat and particle fluxes are mainly caused by electrostatic perturbation, and the contribution of magnetic perturbation is small and negative. The electron heat flux is caused by both electrostatic and magnetic perturbations. The numerical solutions satisfy the entropy balance equation, and the entropy is transferred from ions to electrons through electrostatic and magnetic perturbations. An analysis based on the entropy balance equation shows that the zonal structure is produced by magnetic nonlinearity corresponding to the Maxwell stress in the fluid limit but is weakened by the electrostatic one related to the Reynolds stress. A linear analysis on the standard configuration of LHD plasmas shows the suppression of the ion temperature gradient mode by finite-beta effects and the destabilization of KBM at high beta.

L-mode validation studies of gyrokinetic turbulence simulations via multiscale and multifield turbulence measurements on the DIII-D tokamak

A series of carefully designed experiments on DIII-D have taken advantage of a broad set of turbulence and profile diagnostics to rigorously test gyrokinetic turbulence simulations. In this paper the goals, tools and experiments performed in these validation studies are reviewed and specific examples presented. It is found that predictions of transport and fluctuation levels in the mid-core region (0.4 < ρ < 0.75) are in better agreement with experiment than those in the outer region (ρ ⩾ 0.75) where edge coupling effects may become increasingly important and multiscale simulations may also be necessary. Validation studies such as these are crucial in developing confidence in a first-principles based predictive capability for ITER.

The pinch of cold ions from recycling in the tokamak edge pedestal

We apply the “natural fueling mechanism” [W. Wan, S. E. Parker, Y. Chen, and F. W. Perkins, Phys. Plasmas 17, 040701 (2010)] to the edge pedestal. The natural fueling mechanism is where cold ions naturally pinch radially inward for a heat-flux dominated plasma. It is shown from neoclassical-neutral transport coupled simulations that the recycling neutrals and the associated source ions are colder than the main ions in the edge pedestal. These recycling source ions will pinch radially inward due to microturbulence. Gyrokinetic turbulence simulations indicate that near the top of the pedestal, the pinch velocity of the recycling source ions is much higher than the main ion outgoing flow velocity. The turbulent pinch of the recycling source ions may play a role in the edge pedestal transport and dynamics. The cold ion temperature significantly enhances the pinch velocity of the recycling source ions near to the pedestal top. Neoclassical calculations show a cold ion pinch in the pedestal as well.