Recently, experiments on basic plasma physics issues for solving future problems in fusion energy have been performed on a Large Helical Device. There are several problems to be solved in future devices for fusion energy. Emerging issues in burning plasma are: alpha-channeling (ion heating by alpha particles), turbulence and transport in electron dominant heating helium ash exhaust, reduction of the divertor heat load. To solve these problems, understanding the basic plasma physics of (1) wave–particle interaction through (inverse) Landau damping, (2) characteristics of electron-scale (high- k ) turbulence, (3) ion mixing and the isotope effect, and (4) turbulence spreading and detachment, is necessary. This overview discusses the experimental studies on these issues and turbulent transport in multi-ion plasma and other issues in the appendix.

The Eulerian variational formulation is presented to obtain governing equations of the electromagnetic turbulent gyrokinetic system. A local momentum balance in the system is derived from the invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation by extending the previous work [H. Sugama et al., Phys. Plasmas 28, 022312 (2021)]. Polarization and magnetization due to finite gyroradii and electromagnetic microturbulence are correctly described by the gyrokinetic Poisson equation and Ampère's law which are derived from the variational principle. Also shown is how the momentum balance is influenced by including collisions and external sources. Momentum transport due to collisions and turbulence is represented by a symmetric pressure tensor, which originates in a variational derivative of the Lagrangian with respect to the metric tensor. The relations of the axisymmetry and quasi-axisymmetry of the toroidal background magnetic field to a conservation form of the local momentum balance equation are clarified. In addition, an ensemble-averaged total momentum balance equation is shown to take the conservation form even in the background field with no symmetry when a constraint condition representing the macroscopic Ampère's law is imposed on the background field. Using the WKB representation, the ensemble-averaged pressure tensor due to the microturbulence is expressed in detail and it is verified to reproduce the toroidal momentum transport derived in previous works for axisymmetric systems. The local momentum balance equation and the pressure tensor obtained in this work present a useful reference for elaborate gyrokinetic simulation studies of momentum transport processes.

Microturbulence in magnetic confined plasmas contributes to energy exchange between particles of different species as well as the particle and heat fluxes. Although the effect of turbulent energy exchange has not been considered significant in previous studies, it is anticipated to have a greater impact than collisional energy exchange in low collisional plasmas such as those in future fusion reactors. In this study, gyrokinetic simulations are performed to evaluate the energy exchange due to ion temperature gradient (ITG) turbulence in a tokamak configuration. The energy exchange due to the ITG turbulence mainly consists of the cooling of ions in the ∇B-curvature drift motion and the heating of electrons streaming along a field line. It is found that the ITG turbulence transfers energy from ions to electrons regardless of whether the ions or electrons are hotter, which is in marked contrast to the energy transfer by Coulomb collisions. This implies that the ITG turbulence should be suppressed from the viewpoint of sustaining the high ion temperature required for fusion reactions since it prevents energy transfer from alpha-heated electrons to ions as well as enhancing ion heat transport toward the outside of the reactor. Furthermore, linear and nonlinear simulation analyses confirm the feasibility of quasilinear modeling for predicting the turbulent energy exchange in addition to the particle and heat fluxes.

A one-dimensional Vlasov-Poisson system is considered to elucidate how the information entropies of the probability distribution functions of the electron position and velocity variables evolve in the Landau damping process. Considering the initial condition given by the Maxwellian velocity distribution with the spatial density perturbation in the form of the cosine function of the position, we derive linear and quasilinear analytical solutions that accurately describe both early and late time behaviors of the distribution function and the electric field. The validity of these solutions is confirmed by comparison with numerical simulations based on contour dynamics. Using the quasilinear analytical solution, the time evolutions of the velocity distribution function and its kurtosis indicating deviation from the Gaussian distribution are evaluated with the accuracy of the squared perturbation amplitude. We also determine the time evolutions of the information entropies of the electron position and velocity variables and their mutual information. We further consider Coulomb collisions that relax the state in the late-time limit in the collisionless process to the thermal equilibrium state. In this collisional relaxation process, the mutual information of the position and velocity variables decreases to zero, while the total information entropy of the phase-space distribution function increases by the decrease in the mutual information and demonstrates the validity of Boltzmann's H-theorem.

Isotope effects under the influence of a radial electric field are examined in a helical magnetic field configuration. We perform global gyrokinetic simulations with additional poloidal rotations to estimate quasi-linear heat flux due to ion temperature gradient mode under the mixing length model. In single-ion-species plasmas, the mass number dependency of heat flux agrees with gyro-Bohm scaling in the absence of a radial electric field. Favorable mass number dependencies violating gyro-Bohm scaling are observed in the presence of a global radial electric field or a heavy hydrogen component in multi-ion-species plasmas. The radial electric field and the heavy hydrogen component affect the heat flux through an increase of wavelength as well as mode stabilization. Poloidal Mach number characterizes the transition from unfavorable to favorable mass number dependency under radial electric fields. While the heat flux is independent of mass number for a given poloidal Mach number, the heat flux decreases for higher mass numbers in a given radial electric field. The heat flux is also independent of average mass number in multi-ion-species plasmas because the heavy hydrogen component effectively enhances the light hydrogen heat flux. The present results are potentially relevant to the violation of gyro-Bohm scaling observed in the recent deuterium experiments in the Large Helical Device.

A novel scheme to predict the turbulent transport of ion heat of magnetic confined plasmas is developed by combining mathematical optimization techniques employed in data analysis approaches and first-principle gyrokinetic simulations. Gyrokinetic simulation, as a first-principle approach, is a reliable way to predict turbulent transport. However, in terms of the flux-matching [Candy et al., Phys. Plasmas 16, 060704 (2009)], quantitative transport estimates by gyrokinetic simulations incur extremely heavy computational costs. In order to reduce the costs of quantitative transport prediction based on the gyrokinetic simulations, we develop a scheme with the aid of a reduced transport model. In the scheme, optimization techniques are applied to find relevant input parameters for nonlinear gyrokinetic simulations, which should be performed to obtain relevant transport fluxes and to optimize the reduced transport model for a target plasma. The developed scheme can reduce the numbers of the gyrokinetic simulations to perform the quantitative estimate of the turbulent transport levels and plasma profiles. Utilizing the scheme, the predictions for the turbulent transport can be realized by performing the first-principle simulations once for each radial position.

Transport simulation is performed by integrated code using reduced transport models (Toda S et al 2019 Phys. Plasmas 26 012510) in a kinetic electron condition for turbulent heat transport including the effect of zonal flows in helical plasmas. A reduced model can be formulated for the heat diffusivity using only the linear properties, or can be constructed by considering the expression of the quasilinear flux. These reduced models reproduce nonlinear gyrokinetic simulation results for ion temperature gradient mode turbulence by a linear growth rate and zonal flow decay time. Temperature profiles can be obtained when the turbulent heat transport is evaluated by reduced models at each time step in the evolution of integrated simulation. Computational cost using the reduced models where linear gyrokinetic simulation is performed at each time step in the integrated simulation is about two orders of magnitude lower than that using nonlinear gyrokinetic simulation. Stationary temperature profiles are predicted by simulation, in which, the linear simulation is performed at each time step in the integrated simulation for steady heating power. The density profile and the edge temperature are needed in this simulation.

Expressions of polarization and magnetization in magnetically confined plasmas are derived, which include full expansions in the gyroradius to treat effects of both equilibrium and microscopic electromagnetic turbulence. Using the obtained expressions, densities and flows of particles are related to those of gyrocenters. To the first order in the normalized gyroradius expansion, the mean part of the particle flow is given by the sum of the gyrocenter flow and the magnetization flow, which corresponds to the so-called magnetization law in drift kinetics, while the turbulent part contains the polarization flow as well. Collisions make an additional contribution to the second-order particle flow. The mean particle flux across the magnetic surface is of the second-order, and it contains classical, neoclassical, and turbulent transport processes. The Lagrangian variational principle is used to derive the gyrokinetic Poisson and Ampère equations, which properly include mean and turbulent parts so as to be useful for full-f global electromagnetic gyrokinetic simulations. It is found that the second-order Lagrangian term given by the inner product of the turbulent vector potential and the drift velocity consisting of the curvature drift and the ∇B drift should be retained in order for the derived Ampère equation to correctly include the diamagnetic current, which is necessary especially for the full-f high-beta plasma simulations. The turbulent parts of these gyrokinetic Poisson and Ampère equations are confirmed to agree with the results derived from the WKB representation in earlier works.

Characteristics of the turbulence driven transports are investigated in LHD and W7-X. The gyrokinetic non-linear simulation with identical input gradient shows lower ETG driven transport in W7-X and lower ITG driven transport in LHD. The configuration scan experiments in LHD shows that reduced transport associated with reduction of ion scale turbulence at inwardly shifted configuration, where effective helical ripple is low. The identical experiment between W7-X and LHD with similar density and heating power shows clear different density and temperature profiles. Total transport is lower in W7-X at most of the radial location, however, anomalous contribution is lower in LHD. The reduced ion transport in LHD qualitatively agree with gyrokinetic simulation. The effective helical ripple is not ruling parameter to reduce anomalous transport among stellarator/heliotron configuration.

We assess the magnetic field configuration in modern fusion devices by comparing experiments with the same heating power, between a stellarator and a heliotron. The key role of turbulence is evident in the optimized stellarator, while neoclassical processes largely determine the transport in the heliotron device. Gyrokinetic simulations elucidate the underlying mechanisms promoting stronger ion scale turbulence in the stellarator. Similar plasma performances in these experiments suggests that neoclassical and turbulent transport should both be optimized in next step reactor designs.

The Eulerian variational formulation of the gyrokinetic system with electrostatic turbulence is presented in general spatial coordinates by extending our previous work [H. Sugama et al., Phys. Plasmas 25, 102506 (2018)]. The invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation is used to derive the local momentum balance equation satisfied by the gyrocenter distribution functions and the turbulent potential, which are given as solutions of the governing equations. In the symmetric background magnetic field, the derived local momentum balance equation gives rise to the local momentum conservation law in the direction of symmetry. This derivation is in contrast to the conventional method using the spatial translation in which the asymmetric canonical pressure tensor generally enters the momentum balance equation. In the present study, the variation of the Lagrangian density with respect to the metric tensor is taken to directly obtain the symmetric pressure tensor, which includes the effect of turbulence on the momentum transport. In addition, it is shown in this work how the momentum balance is modified when the collision and/or external source terms are added to the gyrokinetic equation. The results obtained here are considered useful for global gyrokinetic simulations investigating both neoclassical and turbulent transport processes even in general non-axisymmetric toroidal systems.

The improved model collision operator proposed by Sugama et al. [Phys. Plasmas 26, 102108 (2019)], in which the original Sugama collision operator [Phys. Plasmas 16, 112503 (2009)] is extended to recover the friction–flow relation of the linearized Landau (exact) collision operator, is newly implemented in a global full-f gyrokinetic simulation code, GT5D. Neoclassical transport simulations of a single ion species plasma in a circular concentric tokamak are performed over the wide collisionality regime. The improved operator is verified to reproduce the neoclassical thermal diffusivity of Braginskii precisely in the Pfirsch–Schlüter regime, where a friction–flow relation of higher accuracy is required than in the lower collisional regime. In addition, it is found in all collisionality regimes that the overestimation of the neoclassical thermal diffusivity and the magnitude of the parallel flow coefficient observed for the original Sugama operator can be eliminated by using the improved operator, demonstrating that collisional processes described by the exact operator are correctly retained in the improved operator.

The turbulent transport of magnetic confinement plasmas including multi-ion-particle-species in helical systems such as the Large Helical Device (LHD) [Takeiri et al., Nucl. Fusion 57, 102023 (2017)] and their plasma profile sensitivities are investigated by local flux-tube gyrokinetic simulations. In the multi-ion-species plasmas, while the heat transport of each particle species has slightly different sensitivity towards the plasma temperature gradients and the density gradients, there exist quite different dependencies in the particle transport on the radial gradient profiles of the plasma temperatures and densities between each particle species. Furthermore, in the LHD plasma with the carbon impurity hole structure [Ida et al., Plasma Phys. 16, 056111 (2009)], the turbulent particle transport flux of the impurity carbon ion remains radially inward-directed robustly within the wide ranges of radial gradient profiles of the plasma temperatures and densities.

Using transport models, the impacts of trapped electrons on zonal flows and turbulence in helical field configurations are studied. The effect of the trapped electrons on the characteristic quantities of the linear response for zonal flows is investigated for two different field configurations in the Large Helical Device. The turbulent potential fluctuation, zonal flow potential fluctuation and ion energy transport are quickly predicted by the reduced models for which the linear and nonlinear simulation results are used to determine dimensionless parameters related to turbulent saturation levels and typical zonal flow wavenumbers. The effects of zonal flows on the turbulent transport for the case of the kinetic electron response are much smaller than or comparable to those in an adiabatic electron condition for the two different field configurations. It is clarified that the effect of zonal flows on the turbulent transport due to the trapped electrons changes, depending on the field configurations.

The linearized model collision operator for multiple species plasmas given by Sugama et al. [Phys. Plasmas 16, 112503 (2009)] is improved to be properly applicable up to the highly collisional regime. The improved linearized model operator retains the conservation laws of particles, momentum, and energy, and it reproduces the same friction-flow relations as derived by the linearized Landau operator so that this model can be used to correctly evaluate neoclassical transport fluxes in all collisionality regimes. The adjointness relations and Boltzmann's H-theorem are exactly satisfied by the improved operator except in the case of collisions between unlike particle species with unequal temperatures where these relations and H-theorem still hold approximately because there is a large difference between the masses of the two species with significantly different temperatures. Even in the unequal-temperature case, the improved operator can also be modified so as to exactly satisfy the adjointness relations, while it causes the values of the friction coefficients to deviate from those given by the Landau operator. In addition, for application to gyrokinetic simulations of turbulent transport, the improved operator is transformed into the gyrophase-averaged form by keeping the finite gyroradius effect.

The particle and heat transport driven by the ion temperature gradient instability in helical plasmas is investigated by the gyrokinetic analysis taking into account the kinetic electron response. High and low ion temperature plasma cases for the discharge in the Large Helical Device (LHD) are studied. Two types of transport models with a lower computational cost to reproduce the nonlinear gyrokinetic simulation results within allowable errors are presented for application in quick transport analyses. The turbulent electron and ion heat diffusivity models are given in terms of the linear growth rate and the characteristic quantity for the linear response of zonal flows, while the model of the effective particle diffusivity is not obtained for the flattened density profile observed in the LHD. The quasilinear flux model is also shown for the heat transport. The quasilinear flux models for the energy fluxes are found to reproduce the nonlinear simulation results at the accuracy similar to that of the heat diffusivity models. In addition, the quasilinear particle flux model, which is applicable to the transport analysis for LHD plasmas, is constructed. These turbulent reduced models enable coupling to the other simulation in the integrated codes for the LHD.

For the realization of steady-state operation of the burning plasma in future fusion reactor, it is crucial to understand and control the transport process of impurity ions in the core plasma. In a fusion reactor, there are several source of impurity ions: He ash born in the core by D-T reaction, W and Fe from divertor and vacuum vessel wall, etc. Accumulation of heavy impurity ions to the core region should be avoided, since they cool down the core plasma by radiation loss. In ASDEX-U tokamak, it was observed that the central W accumulation was mitigated when a saturated m/n=1/1 kink mode presented in the core [1]. In RFX-mod RFP, plasmas with m/n=1/7 quasi-signlehelicity (QSH) core deformation was found to prevent accumulation of impurities [2]. The helically-deformed state of MHD equilibrium in originally axisymmetric devices appears either as a self-organization process or driven by external magnetic perturbation. It is demonstrated by numerical simulation that such an MHD equilibrium with 3D helical core is possible in tokamak and RFP [3]. Similar mitigation of impurity accumulation is also found in a helical device, LHD. Hollow density profile, or “impurity hole” of carbon impurity, was observed in high-Ti core plasma [4]. Therefore, it is considered that the 3D equilibrium and the transport process in it has some relation with the observed outward impurity transport, of which mechanism is still unknown yet. Concerning the transport process in 3D MHD equilibrium, neoclassical transport has a relatively large impact compared to that in tokamak. Since ion and electron neoclassical particle fluxes in helical plasma are NOT intrinsic ambipolar, spontaneous formation of the ambipolar radial electric field so that the total radial current vanishes is also an important characteristic of the neoclassical transport in a helical plasma. Neoclassical transport theory and numerical simulations have been used to explain the impurity hole phenomenon. However, neoclassical impurity transport in the condition in which impurity hole appears usually opposes to the observation: inward impurity flux is predicted from neoclassical simulations [4]. Therefore, several models have been proposed to explain the impurity hole, such as the contribution of the turbulent transport [6], extended neoclassical transport model which includes the potential variation on flux surface, or the φ1 potential effect [5], etc. However, even such extended models still have not successfully explained the impurity hole theoretically. Drift-kinetic equation for multi-ion species plasmas, especially in 3D configuration, requires large computation resource to evaluate the neoclassical transport. Therefore, to our knowledge, previous studies on the impurity transport in helical plasmas have been carried out using a radially-local approximation model of drift-kinetic equation solver. A possible missing piece to explain the impurity hole is a consideration of the finite radial drift of guiding-center motion by using a global drift-kinetic model. From this point of view, FORTEC-3D [7], which solves the global drift-kinetic equation for single ion species plasma in 3D magnetic configuration, is extended for multi-ion species plasmas. Here, we report the first benchmark of the new code. To treat a multi-ion species plasma, the collision operator in FORTEC-3D is extended for unlike-species collisions. We implement the model of linearized test-particle and field-particle operator proposed by Sugama[8], which possess the adjointness nature of the operator as well as the conservation low in the unlike-species collision, even if the temperature of two species are different. Firstly, these characteristics of the new collision operator is verified. It is demonstrated that the conservation property of particle number, momentum, and energy are satisfied to a rounding error level by using the adaptive field-particle operator and the weight-spreading reduction method, which are also used in the single-species version of FORTEC-3D. Relations between the weight-spreading and Boltzmann’s H-theorem is explained in the presentation. Secondly, we will demonstrate a multi-ion species neoclassical transport simulation in a LHD configuration. Here, the φ1 potential effect is neglected for simplicity. Comparison with DKES/PENTA[9] code, which can solve a radially-local drift-kinetic equation for multi-species plasmas, are shown. Difference in the ambipolar electric field and impurity transport between local and global neoclassical calculations is examined. Since DKES/PENTA can switch the treatment of the parallel momentum transfer in Coulomb collisions, we investigate the impact of momentum balance property on the impurity transport calculation. Benchmark with another global code for helical plasma, GT5D[10], in a LHD configuration will also be presented.

The Eulerian variational principle for the Vlasov-Poisson-Ampère system of equations in a general coordinate system is presented. The invariance of the action integral under an arbitrary spatial coordinate transformation is used to obtain the momentum conservation law and the symmetric pressure in a more direct way than using the translational and rotational symmetries of the system. Next, the Eulerian variational principle is given for the collisionless drift kinetic equation, where particles' phase-space trajectories in given electromagnetic fields are described by Littlejohn's guiding center equations [R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983)]. Then, it is shown that, in comparison with the conventional moment method, the invariance under a general spatial coordinate transformation yields a more convenient way to obtain the momentum balance as a three-dimensional vector equation in which the symmetric pressure tensor, the Lorentz force, and the magnetization current are properly expressed. Furthermore, the Eulerian formulation is presented for the extended drift kinetic system, for which, in addition to the drift kinetic equations for the distribution functions of all particle species, the quasineutrality condition and Ampère's law to determine the self-consistent electromagnetic fields are given. Again, the momentum conservation law for the extended system is derived from the invariance under the general spatial coordinate transformation. Besides, the momentum balances are investigated for the cases where the collision and/or external source terms are added to the Vlasov and drift kinetic equations.

As the finalization of a hydrogen experiment towards the deuterium phase, the exploration of the best performance of hydrogen plasma was intensively performed in the large helical device. High ion and electron temperatures, T-i and T-e, of more than 6 keV were simultaneously achieved by superimposing high-power electron cyclotron resonance heating onneutral beam injection (NBI) heated plasma. Although flattening of the ion temperature profile in the core region was observed during the discharges, one could avoid degradation by increasing the electron density. Another key parameter to present plasma performance is an averaged beta value . The high regime around 4% was extended to an order of magnitude lower than the earlier collisional regime. Impurity behaviour in hydrogen discharges with NBI heating was also classified with a wide range of edge plasma parameters. The existence of a no impurity accumulation regime, where the high performance plasma is maintained with high power heating > 10 MW, was identified. Wide parameter scan experiments suggest that the toroidal rotation and the turbulence are the candidates for expelling impurities from the core region.