Neoclassical Transport Simulation Research Articles

Neoclassical transport simulations with an improved model collision operator

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.

Benchmark of a new multi-ion-species collision operator for [formula omitted] Monte Carlo neoclassical simulation

A numerical method to implement a linearized Coulomb collision operator in the two-weight δf Monte Carlo method for multi-ion-species neoclassical transport simulation is developed. The conservation properties and the self-adjoint property of the operator in the collisions between two particle species with different temperatures are verified. The linearized operator in a δf Monte Carlo code is benchmarked with other two kinetic simulations, a δf continuum gyrokinetic code with the same linearized collision operator and a full-f PIC code with Nanbu collision operator. The benchmark simulations of the equilibration process of plasma flow and temperature fluctuation among several particle species show very good agreement between δf Monte Carlo code and the other two codes. An error in the H-theorem in the two-weight δf Monte Carlo method is found, which is caused by the weight spreading phenomenon inherent in the two-weight δf method. It is demonstrated that the weight averaging method serves to restoring the H-theorem without causing side effect.

Benchmark of the local drift-kinetic models for neoclassical transport simulation in helical plasmas

The benchmarks of the neoclassical transport codes based on the several local drift-kinetic models are reported here. Here, the drift-kinetic models are zero orbit width (ZOW), zero magnetic drift, DKES-like, and global, as classified in Matsuoka et al. [Phys. Plasmas 22, 072511 (2015)]. The magnetic geometries of Helically Symmetric Experiment, Large Helical Device (LHD), and Wendelstein 7-X are employed in the benchmarks. It is found that the assumption of E×B incompressibility causes discrepancy of neoclassical radial flux and parallel flow among the models when E×B is sufficiently large compared to the magnetic drift velocities. For example, Mp≤0.4 where Mp is the poloidal Mach number. On the other hand, when E×B and the magnetic drift velocities are comparable, the tangential magnetic drift, which is included in both the global and ZOW models, fills the role of suppressing unphysical peaking of neoclassical radial-fluxes found in the other local models at Er≃0. In low collisionality plasmas, in particular, the tangential drift effect works well to suppress such unphysical behavior of the radial transport caused in the simulations. It is demonstrated that the ZOW model has the advantage of mitigating the unphysical behavior in the several magnetic geometries, and that it also implements the evaluation of bootstrap current in LHD with the low computation cost compared to the global model.

Impact of nonlinear 3D equilibrium response on edge topology and divertor heat load in Wendelstein 7-X

The impact of the 3D equilibrium response on the plasma edge topology is studied. In Wendelstein 7-X, the island divertor concept is used to assess scenarios for quasi-steady-state operation. However, the boundary islands necessary for the island divertor are strongly susceptible to plasma beta and toroidal current density effects because of the low magnetic shear. The edge magnetic topology for quasi-steady-state operation scenarios is calculated with the HINT-code to study the accompanying changes of the magnetic field structures. Two magnetic configurations have been selected, which had been investigated in self consistent neoclassical transport simulations for low bootstrap current but which use the alternative natural island chains to the standard iota value of 1 (ιb = 5/5, periodicity), namely, at high-iota (ιb = 5/4) and at low-iota (ιb = 5/6). For the high-iota configuration, the boundary islands are robust but the stochasticity around them is enhanced with beta. The addition of toroidal current densities enhances the stochasticity further. The increased stochasticity changes the footprints on in-vessel components with a direct impact on the corresponding heat loads. In the low-iota configuration the boundary islands used for the island divertor are dislocated radially due to the low shear and even show healing effects, i.e. the island width vanishes. In the latter case the plasma changes from divertor to limiter operation. To realize the predicted high-performance quasi-steady-state operation of the transport simulations, further adjustments of the magnetic configuration may be necessary to achieve a proper divertor compatibility of the scenarios.

Effects of magnetic drift tangential to magnetic surfaces on neoclassical transport in non-axisymmetric plasmas

In evaluating neoclassical transport by radially local simulations, the magnetic drift tangential to a flux surface is usually ignored in order to keep the phase-space volume conservation. In this paper, effect of the tangential magnetic drift on the local neoclassical transport is investigated. To retain the effect of the tangential magnetic drift in the local treatment of neoclassical transport, a new local formulation for the drift kinetic simulation is developed. The compressibility of the phase-space volume caused by the tangential magnetic drift is regarded as a source term for the drift kinetic equation, which is solved by using a two-weight δf Monte Carlo method for non-Hamiltonian system [G. Hu and J. A. Krommes, Phys. Plasmas 1, 863 (1994)]. It is demonstrated that the effect of the drift is negligible for the neoclassical transport in tokamaks. In non-axisymmetric systems, however, the tangential magnetic drift substantially changes the dependence of the neoclassical transport on the radial electric field Er. The peaked behavior of the neoclassical radial fluxes around Er = 0 observed in conventional local neoclassical transport simulations is removed by taking the tangential magnetic drift into account.

Physics in the magnetic configuration space of W7-X

The neoclassical confinement and the bootstrap current are analysed in the configuration space of W7-X by self-consistent neoclassical transport simulations. Since the establishment of quasi-stationary operation is the most important goal for W7-X, the analysis concentrates on high-performance discharge scenarios in magnetic configurations which are adjusted so that bootstrap current vanishes, or, alternatively, on scenarios where the bootstrap current can be balanced by strong ECCD. Both scenarios lead to restrictions either in the configuration space or in plasma parameters and ECRH heating scenarios. Furthermore, the flexibility of the magnetic configuration space of W7-X is briefly described with emphasis on other physics topics of interest, for example, ballooning unstable configurations as well as configurations with a magnetic hill which might lead to interchange instability.

Application of an improved control-variate scheme to local neoclassical transport simulations

A new radially local neoclassical transport code is developed based on the radially global two-weight δf Monte Carlo code, FORTEC-3D (Satake et al., 2008). In the collisional two-weight δf method, the variance of weight increases in time due to the so-called weight spreading, which leads to an increasing numerical noise in long time simulations. A new improved control-variate scheme was proposed by Kleiber et al. (2011) to reduce the variance. We investigate the effectiveness and validity of the new control-variate scheme for a practical collisional transport problem in a plasma described by the drift kinetic equation. It is demonstrated that the new scheme reduces the variance of weight by approximately 75% in an axisymmetric magnetic field configuration. The burst-like behavior of a transport observable caused by a large numerical noise can be avoided. As a result, the time evolution of a transport observable can be successfully smoothed.

Simulation of neoclassical transport with the continuum gyrokinetic code COGENT

The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy. Topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.

Full linearized Fokker–Planck collisions in neoclassical transport simulations

The complete linearized Fokker–Planck collision operator has been implemented in the drift-kinetic code NEO (Belli and Candy 2008 Plasma Phys. Control. Fusion 50 095010) for the calculation of neoclassical transport coefficients and flows. A key aspect of this work is the development of a fast numerical algorithm for treatment of the field particle operator. This Eulerian algorithm can accurately treat the disparate velocity scales that arise in the case of multi-species plasmas. Specifically, a Legendre series expansion in ξ (the cosine of the pitch angle) is combined with a novel Laguerre spectral method in energy to ameliorate the rapid numerical precision loss that occurs for traditional Laguerre spectral methods. We demonstrate the superiority of this approach to alternative spectral and finite-element schemes. The physical accuracy and limitations of more commonly used model collision operators, such as the Connor and Hirshman–Sigmar operators, are studied, and the effects on neoclassical impurity poloidal flows and neoclassical transport for experimental parameters are explored.

Neoclassical transport simulations for stellarators

The benchmarking of the thermal neoclassical transport coefficients is described using examples of the Large Helical Device (LHD) and TJ-II stellarators. The thermal coefficients are evaluated by energy convolution of the monoenergetic coefficients obtained by direct interpolation or neural network techniques from the databases precalculated by different codes. The temperature profiles are calculated by a predictive transport code from the energy balance equations with the ambipolar radial electric field estimated from a diffusion equation to guarantee a unique and smooth solution, although several solutions of the ambipolarity condition may exist when root-finding is invoked; the density profiles are fixed. The thermal transport coefficients as well as the ambipolar radial electric field are compared and very reasonable agreement is found for both configurations. Together with an additional W7-X case, these configurations represent very different degrees of neoclassical confinement at low collisionalities. The impact of the neoclassical optimization on the energy confinement time is evaluated and the confinement times for different devices predicted by transport modeling are compared with the standard scaling for stellarators. Finally, all configurations are scaled to the same volume for a direct comparison of the volume-averaged pressure and the neoclassical degree of optimization.

Benchmark test of drift-kinetic and gyrokinetic codes through neoclassical transport simulations

Two simulation codes that solve the drift-kinetic or gyrokinetic equation in toroidal plasmas are benchmarked by comparing the simulation results of neoclassical transport. The two codes are the drift-kinetic δf Monte Carlo code (FORTEC-3D) and the gyrokinetic full- f Vlasov code (GT5D), both of which solve radially-global, five-dimensional kinetic equation with including the linear Fokker–Planck collision operator. In a tokamak configuration, neoclassical radial heat flux and the parallel flow relation, which relates the parallel mean flow with radial electric field and temperature gradient, are compared between these two codes, and their results are also compared with the local neoclassical transport theory. It is found that the simulation results of the two codes coincide very well in a wide rage of plasma collisionality parameter ν ∗ = 0.01 ∼ 10 and also agree with the theoretical estimations. The time evolution of radial electric field and particle flux, and the radial profile of the geodesic acoustic mode frequency also coincide very well. These facts guarantee the capability of GT5D to simulate plasma turbulence transport with including proper neoclassical effects of collisional diffusion and equilibrium radial electric field.

Development of a Non-Local Neoclassical Transport Code for Helical Configurations

The progress in a 3-dimensional, non-local neoclassical transport simulation code “FORTEC-3D” is described. The main purpose of the code is to solve the drift-kinetic equation in general a 3-dimensional configuration using the δf Monte Carlo method, and to calculate neoclassical fluxes and the time evolution of the ambipolar radial electric field simultaneously. This article explains new numerical schemes adopted in FORTEC-3D in order to overcome numerical problems, which happen especially in the cases where the bifurcation of radial electric field occurs. Examples of test simulation for an LHD magnetic field configuration with a bifurcated electric field are also shown. With improved numerical schemes, FORTEC-3D can calculate neoclassical fluxes and trace the time evolution stably for several ion collision times, which is sufficiently long to observe GAM damping and formation of the ambipolar electric field.

Non-Local Simulation of the Formation of Neoclassical Ambipolar Electric Field in Non-Axisymmetric Configurations

Neoclassical transport simulation code (FORTEC-3D) applicable to non-axisymmetric configurations is developed. A new hybrid simulation model in which ion transport is solved by using the δf Monte-Carlo method including the finite-orbit-width effects, while electron transport is solved by a reduced ripple-averaged kinetic equation, is adopted. This model makes it possible to simulate the dynamism of non-local transport phenomena with a self-consistently developing radial electric field within an allowable computation time. Time evolution of a radial electric field in LHD plasma is simulated in the full volume of the confinement region, and the finite-orbit-width effect of neoclassical transport is found to make the negative ambipolar electric field larger than that predicted by local transport theory.

Non-local neoclassical transport simulation of geodesic acoustic mode

Neoclassical transport simulation code (FORTEC-3D) applicable to both axisymmetric and non-axisymmetric configurations is developed to investigate non-local effects on neoclassical transport phenomena. The time evolution of the radial electric field is simulated in the full volume of the confinement region of tokamak and helical model plasmas. It is found that the damping rate of the geodesic-acoustic-mode (GAM) oscillation becomes faster than that predicted from a single-surface transport analysis. The time evolution of the radial electric field towards the ambipolar state shows a non-local behaviour, which indicates a coupling of GAM oscillation between the neighbouring two flux surfaces because of the finite-orbit-width effect.