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Gyrokinetic simulations are required for the quantitative calculation of fluxes due to turbulence, which dominates over other transport mechanisms in tokamaks. However, nonlinear gyrokinetic simulations are computationally expensive. A multimodal convolutional neural network model that reads images and values generated by nonlinear gyrokinetic simulations and predicts electrostatic turbulent heat fluxes was developed to support efficient runs. The model was extended to account for squared electrostatic potential fluctuations, which are proportional to the fluxes in the quasilinear model, as well as images containing fluctuating electron and ion distribution functions and fluctuating electrostatic potentials in wavenumber space. This multimodal model can predict the time and electron and ion turbulent heat fluxes corresponding to the input data. The model trained on the Cyclone base case data successfully predicted times and fluxes not only for its test data, but also for the completely different and unknown JT-60U case, with high accuracy. The predictive performance of the model depended on the similarity of the linear stability of the case used to train the model to the case being predicted.

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Plasma β dependence of electromagnetic turbulent transport is investigated by means of gyrokinetic simulations with self-consistent change of the equilibrium magnetic field. It is found that energy transport due to ion-temperature-gradient (ITG) driven turbulence does not decrease with increasing β; that is, the ion energy diffusivity does not decrease, and the electron energy diffusivity increases with β. This is because magnetic fluctuations are significantly influenced by the background magnetic field structure change with β by the Pfirsch-Schluter current. The magnetic field change weakens the suppression effect of magnetic perturbations on the growth of the ITG mode, and it also suppresses nonlinear zonal flow production. The influence of the magnetic field change is significant as the global magnetic shear increases.

Effects of kinetic ions and electromagnetic fluctuations on slab electron-temperature-gradient (ETG) driven turbulence are investigated by means of gyrokinetic simulations covering scales from electron gyroradius to ion gyroradius and electron skin depth. Linear growth rates of electrostatic ETG modes are enhanced by the ion polarization at ion gyroradius scale. Nonlinear simulations show that this low-poloidal-wavenumber instability induces ion-scale eddies which cause heat transport higher than that in the adiabatic ion model. In finite-β plasma, electromagnetic fluctuations stabilize the low-wavenumber modes, and suppress the turbulent heat transport. Additionally, electromagnetic effects also weaken zonal flow generation, which slightly enhances turbulent transport than that in the electrostatic adiabatic ion model.

This article discusses triad transfer analysis via quadratic nonlinearity. To avoid fictitious interactions, symmetrization of the triad transfer is reviewed, including arbitrary orthogonal decomposition and coarse graining. The directional representation of the symmetrized triad transfer is proposed by minimizing the number of edges in a network graph of triad interactions with keeping the energy consistency. The directional representation simplifies visualization and allows us to reduce the energy transfer into a one-to-one relation, while eliminating fictitious interactions in non-symmetrized triad transfer functions. Energy transfer processes among plasma turbulent fluctuations that decompose by the singular value decomposition are analyzed as an application. A network graph visualization clearly demonstrates the importance of symmetrization and the consistency between the symmetrized triad transfer and its directional representation.

Phase space structures are extracted from the time series of five dimensional distribution function data computed by the flux-driven full-f gyrokinetic code GT5D. Principal component analysis (PCA) is applied to reduce the dimensionality and the size of the data. Phase space bases in ( φ , v ∥ , w ) and the corresponding spatial coefficients (poloidal cross section) are constructed by PCA, where φ , v ∥, and w, respectively, mean the toroidal angle, the parallel velocity, and the perpendicular velocity. It is shown that 83% of the variance of the original five dimensional distribution function can be expressed with 64 principal components, i.e., the compression of the degrees of freedom from 1.3 × 1 0 12 to 1.4 × 10 9. One of the important findings—resulting from the detailed analysis of the contribution of each principal component to the energy flux—deals with avalanche events, which are found to be mostly driven by coherent structures in the phase space, indicating the key role of resonant particles. Another advantage of the proposed analysis is the decoupling of 6D (1D time and 5D phase space) data into the combinations of 3D data which are visible to the human eye.

The transfer rate of the entropy variable in the wavenumber space perpendicular to the equilibrium magnetic field is evaluated by means of the gyrokinetic simulations of the toroidal ion temperature gradient turbulence. Fluctuations of the perturbed gyrocenter distribution function of the linearly unstable mode are transferred to those with higher radial wavenumbers through interactions with zonal flows driven by turbulence. The entropy transfer rate coincides with the zonal flow shearing rate in cases with weak ion temperature gradient where the zonal flows effectively regulate the turbulent transport. The obtained result verifies a correlation between the entropy transfer and the zonal flow shearing processes.

Numerical techniques for parallel dynamics in electromagnetic gyrokinetic simulations are introduced to regulate unphysical grid-size oscillations in the field-aligned coordinate. It is found that a fixed boundary condition and the nonlinear mode coupling in the field-aligned coordinate, as well as numerical errors of non-dissipative finite difference methods, produce fluctuations with high parallel wave numbers. The theoretical and numerical analyses demonstrate that an outflow boundary condition and a low-pass filter efficiently remove the numerical oscillations, providing small but acceptable errors of the entropy variables. The new method is advantageous for quantitative evaluation of the entropy balance that is required for obtaining a steady state in gyrokinetic turbulence.

A semi-Lagrangian scheme is applied for the first time to computations of charged particle motions along magnetic field lines, to numerically solve the δf gyrokinetic equations in a flux tube geometry. This new solver adopted in the gyrokinetic Vlasov simulations has an advantage over the conventional Eulerian codes in calculating the parallel dynamics, because semi-Lagrangian schemes are free of the Courant-Friedrichs-Lewy (CFL) condition that restricts the time step size. A study of the accuracy of the parallel motion simulations reveals that numerical errors mainly stem from spatial (not temporal) discretization for realistic values of the grid spacing and time step, and it demonstrates the advantage of the semi-Lagrangian scheme. This novel numerical method is successfully applied to linear gyrokinetic simulations of the ion temperature gradient instability, where time steps larger than those restricted by the CFL condition can be employed.

Benchmark between electromagnetic gyrokinetic codes GKV and GENE based on the flux tube model has been carried out by employing the high performance tokamak experiment data. The constructed flux coordinates, linear dispersion relation, and nonlinear turbulent transport show good agreements. Utilizing these codes, linear mode structures of micro-tearing modes have been examined. It is found that the current density is highly localized in the radial direction and peaks on the inner board side of the torus, while the broaden O-shape structure is observed on the torus outer board side. This broadening originates from the magnetic drift, that is, the finite orbit width of passing particles.

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 $\mathbit{E}\ifmmode\times\else\texttimes\fi{}\mathbit{B}$ 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.

Multi-scale plasma turbulence including electron and ion temperature gradient (ETG/ITG) modes has been investigated by means of electromagnetic gyrokinetic simulations. Triad transfer analyses on nonlinear mode coupling reveal cross-scale interactions between electron and ion scales. One of the interactions is suppression of electron-scale turbulence by ion-scale turbulence, where ITG-driven short-wavelength eddies act like shear flows and suppress ETG turbulence. Another cross-scale interaction is enhancement of ion-scale turbulence in the presence of electron-scale turbulence. This is caused via short-wavelength zonal flows, which are created by the response of passing kinetic electrons in ITG, suppress ITG by their shearing, and are damped by ETG turbulence. In both cases, sub-ion-scale structures between electron and ion scales play important roles in the cross-scale interactions.

Self-organization in the slab electron temperature gradient driven (ETG) turbulence is investigated based on gyrokinetic simulations and the Hasegawa-Mima (HM) equation. The scale and the anisotropy of self-organized turbulent structures vary depending on the Rhines scale and the characteristic scale given by the adiabatic response term in the HM equation. The former is determined by competition between the linear wave dispersion and the nonlinear turbulent cascade, while the latter is given as the scale, at which the turbulent cascade is impeded. These scales are controlled by plasma parameters such as the density and temperature gradient, and the temperature ratio of ion to electron. It is found that depending on the plasma parameters, the ETG turbulence shows either isotropic turbulence or zonal flows, which give significantly different transport levels. Although the modulational instability excites zonal modes regardless of the plasma parameters, the final turbulent structure is determined by the self...

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.

Quantitative validation studies of flux-tube gyrokinetic Vlasov simulations on ion and electron heat transport are carried out for the JT-60U tokamak experiment. The ion temperature gradient (ITG) and/or trapped electron modes (TEM) driven turbulent transport and zonal flow generations are investigated for an L-mode plasma in the local turbulence limit with a sufficiently small normalized ion thermal gyroradius and weak mean radial electric fields. Nonlinear turbulence simulations by the GKV code successfully reproduce radial profiles of the ion and electron energy fluxes in the core region. The numerical results show that the TEM-driven zonal flow generation in the outer region is more significant than that in the core region with ITG- and ITG–TEM-dominated turbulence, leading to moderate transport shortfall of the ion energy flux. Error levels in the prediction of the ion and electron temperature gradient profiles in the core region are estimated as less than ± 30 % , based on a multiple flux matching technique, where the simulated ion and electron energy fluxes are simultaneously matched to the experimental values.

Gyrokinetic simulations of electromagnetic turbulence in magnetically confined torus plasmas including tokamak and heliotron/stellarator are reviewed. Numerical simulation of turbulence in finite beta plasmas is an important task for predicting the performance of fusion reactors and a great challenge in computational science due to multiple spatio-temporal scales related to electromagnetic ion and electron dynamics. The simulation becomes further challenging in non-axisymmetric plasmas. In finite beta plasmas, magnetic perturbation appears and influences some key mechanisms of turbulent transport, which include linear instability and zonal flow production. Linear analysis shows that the ion-temperature gradient (ITG) instability, which is essentially an electrostatic instability, is unstable at low beta and its growth rate is reduced by magnetic field line bending at finite beta. On the other hand, the kinetic ballooning mode (KBM), which is an electromagnetic instability, is destabilized at high beta. In addition, trapped electron modes (TEMs), electron temperature gradient (ETG) modes, and micro-tearing modes (MTMs) can be destabilized. These instabilities are classified into two categories: ballooning parity and tearing parity modes. These parities are mixed by nonlinear interactions, so that, for instance, the ITG mode excites tearing parity modes. In the nonlinear evolution, the zonal flow shear acts to regulate the ITG driven turbulence at low beta. On the other hand, at finite beta, interplay between the turbulence and zonal flows becomes complicated because the production of zonal flow is influenced by the finite beta effects. When the zonal flows are too weak, turbulence continues to grow beyond a physically relevant level of saturation in finite-beta tokamaks. Nonlinear mode coupling to stable modes can play a role in the saturation of finite beta ITG mode and KBM. Since there is a quadratic conserved quantity, evaluating nonlinear transfer of the conserved quantity from unstable modes to stable modes is useful for understanding the saturation mechanism of turbulence.

Turbulent transport in a high ion temperature discharge of the Large Helical Device (LHD) is investigated by means of electromagnetic gyrokinetic simulations, which include kinetic electrons, magnetic perturbations, and full geometrical effects. Including kinetic electrons enables us to firstly evaluate the particle and the electron heat fluxes caused by turbulence in LHD plasmas. It is found that the electron energy transport reproduces the experimental result, and that the particle flux is negative. The contribution of magnetic perturbation to the transport is small because of very low beta. The turbulence is driven by the ion temperature gradient instability, and the effect of kinetic electrons enhances the growth rate larger than that from the adiabatic electron calculation. The ion energy flux is larger than that observed in the experiment, while the flux is close to the experimental observation when the temperature gradient is reduced 20% in the simulation. This significant sensitivity of the energy flux implies that the profile in the experiment is close to the critical temperature gradient. The critical gradient for turbulent energy flux is similar to that for the linear instability, i.e., the Dimits shift is small. This is because the zonal flow in the LHD is weaker than that in tokamaks.

Plasma turbulence accompanied with fluctuations of the distribution function and the electromagnetic fields develops on the phase space composed of the configuration space and the velocity space. Detailed structures of the distribution function in magnetic fusion plasmas are investigated by means of gyrokinetic simulations performed on massively parallel supercomputers. The gyrokinetic simulations of drift wave turbulence have demonstrated entropy transfer in the phase space, zonal flow enhancement by helical fields and the resultant transport reduction. The state-of-the-art high performance computing is utilized for a multi-scale turbulence simulation covering ion- and electron-scales and for a global-scale simulation of turbulent transport in a sub-ITER sized plasma.

A saturation mechanism for microturbulence in a regime of weak zonal flow generation is investigated by means of electromagnetic gyrokinetic simulations. The study identifies a new saturation process of the kinetic ballooning mode (KBM) turbulence originating from the spatial structure of the KBM instabilities in a finite-beta Large Helical Device (LHD) plasma. Specifically, the most unstable KBM in LHD has an inclined mode structure with respect to the mid-plane of a torus, i.e., it has a finite radial wave-number in flux tube coordinates, in contrast to KBMs in tokamaks as well as ion-temperature gradient modes in tokamaks and helical systems. The simulations reveal that the growth of KBMs in LHD is saturated by nonlinear interactions of oppositely inclined convection cells through mutual shearing as well as by the zonal flow. The saturation mechanism is quantitatively investigated by analysis of the nonlinear entropy transfer that shows not only the mutual shearing but also a self-interaction with an elongated mode structure along the magnetic field line.

Electromagnetic turbulence driven by kinetic ballooning modes (KBMs) in high-β plasma is investigated based on the local gyrokinetic model. Analysis of turbulent fluxes, norms, and phases of fluctuations shows that KBM turbulence gives narrower spectra and smaller phase factors than those in ion-temperature-gradient (ITG)-driven turbulence. This leads to the smaller transport fluxes in KBM turbulence than those in ITG turbulence even when they have similar linear growth rates. From the analysis of the entropy balance relation, it is found that the entropy transfer from ions to electrons through the field-particle interactions mainly drives electron perturbations, which creates radial twisted modes by rapid parallel motions of electrons in a sheared magnetic geometry. The nonlinear coupling between the dominant unstable mode and its twisted modes is important for the saturation of KBM turbulence, in contrast to the importance of zonal flow shearing in ITG turbulence. The coupling depends on the flux-tube domain with the one-poloidal-turn parallel length and on the torus periodicity constraint.

The kinetic ballooning mode (KBM) turbulence in Tokamak plasma is investigated by electromagnetic gyrokinetic simulations. From the entropy balance analysis, it is revealed that the field-particle interactions transfer a significant fraction of the ion entropy produced by the instability to electrons. Then, the produced ion entropy balances to the sum of the ion and electron dissipations at the saturation of the KBM instability growth, in contrast to ITG turbulence where ion entropy production mostly balances to ion dissipation.

A local gyrokinetic Vlasov simulation code GKV is extended to incorporate realistic tokamak equilibria including up-down asymmetry, which are produced by a free-boundary 2D Grad-Shafranov equation solver MEUDAS. By using a newly developed interface code IGS, two dimensional rectangular equilibrium data from MEUDAS is converted to straight-field-line flux coordinates such as Hamada, Boozer, and axisymmetric coordinates, which are useful for gyrokinetic micro-instability and turbulent transport analyses. The developed codes have been verified by a cross-code benchmark test using Cyclone-base-case like MHD equilibrium, where good agreement in the dispersion relation of ion temperature gradient (ITG) driven mode has been confirmed. The extended GKV is applied to two types of shaped plasmas expected in JT-60SA tokamak devices, i.e., ITER-like and highly-shaped plasmas, and ITG-mode stability and residual zonal-flow level are investigated. Through the detailed comparisons, more favorable stability properties against the ITG mode are revealed for the highly-shaped case, where the lower ITG-mode growth rate and higher residual zonal-flow levels compared to the ITER-like case are identified. c © 2014 The Japan Society of Plasma Science and Nuclear Fusion Research

One of the important phenomena in magnetically-confined fusion plasma is plasma turbulence, which causes particle and heat transport and degrades plasma confinement. To address multi-scale turbulence including temporal and spatial scales of electrons and ions, we extend our gyrokinetic Vlasov simulation code GKV to run efficiently on peta-scale supercomputers. A key numerical technique is the parallel Fast Fourier Transform (FFT) required for parallel spectral calculations, where masking of the cost of inter-node transpose communications is essential to improve strong scaling. To mask communication costs, computation-communication overlap techniques are applied for FFTs and transpose with the help of the hybrid parallelization of message passing interface and open multi-processing. Integrated overlaps including whole spectral calculation procedures show better scaling than simple overlaps of FFTs and transpose. The masking of communication costs significantly improves strong scaling of the GKV code, and makes substantial speed-up toward multi-scale turbulence simulations.

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.

Effects of time-varying sheared E×B flow on turbulence driven by slab ion temperature gradient instabilities are investigated by means of Landau fluid simulation. Here, the E×B flow, which consists of stationary and time-periodic oscillatory parts, is externally imposed to the turbulence. The dependence on the amplitude and frequency of E×B flow is examined in the case that the amplitude of oscillatory part is the same or less than that of stationary part. The ion heat transport caused by turbulence oscillates with the same period as the E×B flow and the time-averaged transport coefficient is larger than the coefficient which is evaluated without the oscillatory part. The time-averaged coefficient is maximized when the amplitude of oscillatory part is equal to that of stationary part. As the frequency of E×B flow increases, the time-averaged coefficient decreases and is close to the coefficient which is evaluated without the oscillatory part. This mechanism is explained by introducing a kind of the logist...