Neoclassical Code Research Articles

MONKES: a fast neoclassical code for the evaluation of monoenergetic transport coefficients in stellarator plasmas

MONKES is a new neoclassical code for the evaluation of monoenergetic transport coefficients in stellarators. By means of a convergence study and benchmarks with other codes, it is shown that MONKES is accurate and efficient. The combination of spectral discretization in spatial and velocity coordinates with block sparsity allows MONKES to compute monoenergetic coefficients at low collisionality, in a single core, in approximately one minute. MONKES is sufficiently fast to be integrated into stellarator optimization codes for direct optimization of the bootstrap current and to be included in predictive transport suites. The code and data from this paper are available at https://github.com/JavierEscoto/MONKES/.

How to evaluate neoclassical transport coefficients by a single δf simulation

The δf method is extended to calculate all possible components of neoclassical transport matrices, including non-diagonal components and interspecies components of any species combinations in an arbitrary magnetic field configuration, by a single simulation. The method is implemented in a global neoclassical code FORTEC-3D and applied to cases of multi-ion species plasmas in a tokamak and a helical configuration, respectively. The results are benchmarked against analytic solutions and local simulation results. It is found that the results obtained with the newly developed method provide reliable solutions that satisfy the Onsager symmetry relation as well as the intrinsic ambipolarity in a tokamak configuration.

Finite orbit width effects in large aspect ratio stellarators

New orbit-averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity-space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS (Velasco et al., J. Comput. Phys., vol. 418, 2020, 109512; Velasco et al., Nucl. Fusion, vol. 61, 2021, 116013). The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt {\nu }$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt {\nu }$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by Calvo et al. (Plasma Phys. Control. Fusion, vol. 59, 2017, 055014). In these stellarators, the $\sqrt {\nu }$ regime exists in the collisionality interval $(a/R) \ln (R/a) \ll \nu _{ii} R a/\rho _i v_{ti} \ll R/a$ . In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt {\nu }$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$ : $\delta ^{2} |\ln \delta | \ll \nu _{ii} R a/\rho _i v_{ti} \ll 1$ . Here, $\nu _{ii}$ is the ion–ion collision frequency, $\rho _i$ and $v_{ti}$ are the ion gyroradius and thermal speed, and $a$ and $R$ are the minor and major radii.

Fast simulations for large aspect ratio stellarators with the neoclassical code KNOSOS

In this work, a new version of KNOSOS is presented. KNOSOS is a low-collisionality radially-local, bounce-averaged neoclassical code that is extremely fast, and at the same time, includes physical effects often neglected by more standard codes: the component of the magnetic drift that is tangent to the flux-surface and the variation of the electrostatic potential on the flux-surface. An earlier version of the code could only describe configurations that were sufficiently optimized with respect to neoclassical transport. KNOSOS can now be applied to any large aspect ratio stellarator, and its performance is demonstrated by means of detailed simulations in the configuration space of Wendelstein 7-X.

Radial electric field and density fluctuations measured by Doppler reflectometry during the post-pellet enhanced confinement phase in W7-X

Radial profiles of density fluctuations and the radial electric field, E r, have been measured using Doppler reflectometry during the post-pellet enhanced confinement phase achieved, under different heating power levels and magnetic configurations, during the 2018 W7-X experimental campaign. A pronounced E r-well is measured with local values as high as −40 kV m−1 in the radial range ρ ∼ 0.7–0.8 during the post-pellet enhanced confinement phase. The maximum E r intensity scales with both the plasma density and electron cyclotron heating power level, following a similar trend to the plasma energy content. A good agreement is found when the experimental E r profiles are compared to simulations carried out using the neoclassical codes, the drift kinetic equation solver (DKES) and kinetic orbit-averaging solver for stellarators (KNOSOS). The density fluctuation level decreases from the plasma edge toward the plasma core and the drop is more pronounced in the post-pellet enhanced confinement phase than in reference gas-fuelled plasmas. Besides, in the post-pellet phase, the density fluctuation level is lower in the high iota magnetic configuration than in the standard one. To determine whether this difference is related to the differences in the plasma profiles or to the stability properties of the two configurations, gyrokinetic simulations have been carried out using the codes stella and EUTERPE. The simulation results point to the plasma profile evolution after the pellet injection and the stabilization effect of the radial electric field profile as the dominant players in the stabilization of the plasma turbulence.

Global calculation of neoclassical impurity transport including the variation of electrostatic potential

Recently, the validity range of the approximations commonly used in neoclassical calculation has been reconsidered. One of the primary motivations behind this trend is observation of an impurity hole in LHD (Large Helical Device), i.e. the formation of an extremely hollow density profile of an impurity ion species, such as carbon $\text{C}^{6+}$ , in the plasma core region where a negative radial electric field ( $E_{r}$ ) is expected to exist. Recent studies have shown that the variation of electrostatic potential on the flux surface, $\unicode[STIX]{x1D6F7}_{1}$ , has significant impact on neoclassical impurity transport. Nevertheless, the effect of $\unicode[STIX]{x1D6F7}_{1}$ has been studied with radially local codes and the necessity of global calculation has been suggested. Thus, we have extended a global neoclassical code, FORTEC-3D, to simulate impurity transport in an impurity hole plasma including $\unicode[STIX]{x1D6F7}_{1}$ globally. Independently of the $\unicode[STIX]{x1D6F7}_{1}$ effect, an electron root of the ambipolar condition for the impurity hole plasma has been found by global simulation. Hence, we have considered two different cases, each with a positive (global) and a negative (local) solution of the ambipolar condition, respectively. Our result provides another support that $\unicode[STIX]{x1D6F7}_{1}$ has non-negligible impact on impurity transport. However, for the ion-root case, the radial $\text{C}^{6+}$ flux is driven further inwardly by $\unicode[STIX]{x1D6F7}_{1}$ . For the electron-root case, on the other hand, the radial particle $\text{C}^{6+}$ flux is outwardly enhanced by $\unicode[STIX]{x1D6F7}_{1}$ . These results indicate that how $\unicode[STIX]{x1D6F7}_{1}$ affects the radial particle transport crucially depends on the profile of the ambipolar- $E_{r}$ , which is found to be susceptible to $\unicode[STIX]{x1D6F7}_{1}$ itself and the global effects.

Cross-verification of neoclassical transport solutions from XGCa against NEO

Radial neoclassical transport from the global gyrokinetic code XGC has been cross-verified against the local drift-kinetic neoclassical code NEO in NEO's local regime using a Grad-Shafranov equilibrium with a circular boundary. XGC shows excellent agreement with NEO, except for a difference of approximately 10% in the ion heat flux. Close investigation reveals that this difference is due to the simplification in NEO to use the same Coulomb logarithm ln Λ to represent all particle species, while XGC uses different formulas for collisions between different species [J. D. Huba, NRL Plasma Formulary (Naval Research Laboratory, Washington, DC, 2018)]. When XGC uses NEO's Coulomb logarithm, excellent agreement in the ion heat flux is recovered. The present work not only crossverifies the equation solvers in two codes, but also verifies that the gyrokinetic and the drift-kinetic equations agree and that the linearized and nonlinear Coulomb collision operators agree in the mild gradient, local regime.

Pressure balance in a lower collisionality, attached tokamak scrape-off layer

Previously the gyrokinetic neoclassical code XGCa found that the pressure balance (or momentum balance) in the diverted scrape-off layer (SOL) does not follow that from the fluid description based on the Chew–Goldberger–Low (CGL) theory. In this paper a gyrocenter parallel momentum moment equation is derived for pressure balance (or momentum balance) in a tokamak SOL, for use in interpreting this difference. The new gyrocenter-fluid pressure balance equation allows identifying from the XGCa code results which physical processes dominate the setting of pressure variation in the scrape-off layer. This pressure balance equation is applied to the simulation of a DIII-D H-mode discharge, with a lower ion collisionality in the SOL, and the Coulomb and atomic collisions are not strong enough to yield a detached divertor plasma. It is found that the total pressure balance is much better matched using the gyrocenter parallel momentum moment equation. Electrons are shown to be dominantly adiabatic, while ions have multiple contributions to pressure balance, including terms originating from particle drifts. These results show that in strong gradient, low collisionality regions of the SOL, the typical fluid reductions miss important effects captured in the gyrokinetic model.

Turbulence and perpendicular plasma flow asymmetries measured at TJ-II plasmas

Dedicated experiments have been carried out for a systematic comparison of turbulence wavenumber spectra and perpendicular rotation velocity measured at poloidally separated positions on the same flux-surface in the stellarator TJ-II. The rationale behind this study is twofold, namely, validation of the spatial localization of instabilities predicted by gyrokinetic simulations in stellarators and validation of the electrostatic potential variation on the flux surface as calculated by neoclassical codes and its possible impact on the radial electric field. Perpendicular wavenumber spectra and perpendicular rotation velocity profiles have been measured using Doppler reflectometry in two plasma regions poloidally separated as both positive and negative probing beam angles with respect to normal incidence can be selected. A systematic comparison has been carried out showing differences in the perpendicular wavenumber spectrum measured at poloidally separated positions on the same flux-surface, that depend on plasma density, heating conditions and magnetic configuration. The asymmetry found in the standard magnetic configuration under some plasmas conditions, reverses in the high iota configuration. The different intensity in the density fluctuation spectra can be related to the poloidal localization of instabilities found in gyrokinetic simulations. Differences in the radial electric field profile are also found that could be explained to be due to on-surface plasma potential variations.

Electrostatic potential variations on stellarator magnetic surfaces in low collisionality regimes

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by$\tilde{\unicode[STIX]{x1D711}}$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which$\tilde{\unicode[STIX]{x1D711}}$is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio and the structure of$\tilde{\unicode[STIX]{x1D711}}$on the magnetic surface are analytically derived in the$1/\unicode[STIX]{x1D708}$,$\sqrt{\unicode[STIX]{x1D708}}$and superbanana-plateau regimes of stellarators close to omnigeneity; i.e. stellarators that have been optimized for neoclassical transport. It is found that the largest$\tilde{\unicode[STIX]{x1D711}}$that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigenous configuration, the analytical results are verified and illustrated with calculations by the codeKNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.

Core radial electric field and transport in Wendelstein 7-X plasmas

The results from the investigation of neoclassical core transport and the role of the radial electric field profile (Er) in the first operational phase of the Wendelstein 7-X (W7-X) stellarator are presented. In stellarator plasmas, the details of the Er profile are expected to have a strong effect on both the particle and heat fluxes. Investigation of the radial electric field is important in understanding neoclassical transport and in validation of neoclassical calculations. The radial electric field is closely related to the perpendicular plasma flow (u⊥) through the force balance equation. This allows the radial electric field to be inferred from measurements of the perpendicular flow velocity, which can be measured using the x-ray imaging crystal spectrometer and correlation reflectometry diagnostics. Large changes in the perpendicular rotation, on the order of Δu⊥∼ 5 km/s (ΔEr ∼ 12 kV/m), have been observed within a set of experiments where the heating power was stepped down from 2 MW to 0.6 MW. These experiments are examined in detail to explore the relationship between heating power temperature, and density profiles and the radial electric field. Finally, the inferred Er profiles are compared to initial neoclassical calculations based on measured plasma profiles. The results from several neoclassical codes, sfincs, fortec-3d, and dkes, are compared both with each other and the measurements. These comparisons show good agreement, giving confidence in the applicability of the neoclassical calculations to the W7-X configuration.

High Z neoclassical transport: Application and limitation of analytical formulae for modelling JET experimental parameters

Heavy impurities, such as tungsten (W), can exhibit strongly poloidally asymmetric density profiles in rotating or radio frequency heated plasmas. In the metallic environment of JET, the poloidal asymmetry of tungsten enhances its neoclassical transport up to an order of magnitude, so that neoclassical convection dominates over turbulent transport in the core. Accounting for asymmetries in neoclassical transport is hence necessary in the integrated modeling framework. The neoclassical drift kinetic code, NEO [E. Belli and J. Candy, Plasma Phys. Controlled Fusion P50, 095010 (2008)], includes the impact of poloidal asymmetries on W transport. However, the computational cost required to run NEO slows down significantly integrated modeling. A previous analytical formulation to describe heavy impurity neoclassical transport in the presence of poloidal asymmetries in specific collisional regimes [C. Angioni and P. Helander, Plasma Phys. Controlled Fusion 56, 124001 (2014)] is compared in this work to numerical results from NEO. Within the domain of validity of the formula, the factor for reducing the temperature screening due to poloidal asymmetries had to be empirically adjusted. After adjustment, the modified formula can reproduce NEO results outside of its definition domain, with some limitations: When main ions are in the banana regime, the formula reproduces NEO results whatever the collisionality regime of impurities, provided that the poloidal asymmetry is not too large. However, for very strong poloidal asymmetries, agreement requires impurities in the Pfirsch-Schlüter regime. Within the JETTO integrated transport code, the analytical formula combined with the poloidally symmetric neoclassical code NCLASS [W. A. Houlberg et al., Phys. Plasmas 4, 3230 (1997)] predicts the same tungsten profile as NEO in certain cases, while saving a factor of one thousand in computer time, which can be useful in scoping studies. The parametric dependencies of the temperature screening reduction due to poloidal asymmetries would need to be better characterised for this faster model to be extended to a more general applicability.

Measurement of the complete core plasma flow across the LOC–SOC transition at ASDEX Upgrade

A newly installed core charge exchange recombination spectroscopy (CXRS) diagnostic at ASDEX Upgrade (AUG) enables the evaluation of the core poloidal rotation () through the inboard-outboard asymmetry of the toroidal rotation with an accuracy of 0.5 to 1 km . Using this technique, the total plasma flow has been measured in Ohmic L-mode plasmas across the transition from the linear to saturated ohmic confinement (LOC–SOC) regimes. The core poloidal rotation of the plasma around mid-radius is found to be always in the ion diamagnetic direction, in disagreement with neoclassical (NC) predictions. The edge rotation is found to be electron-directed and consistent with NC codes. This measurement provides as well the missing ingredient to evaluate the core velocity () from data only, which can then be compared to measurements of the perpendicular velocity of the turbulent fluctuations () to gain information on the turbulent phase velocity (). The non neoclassical from CXRS leads to good agreement between and indicating that is small and at similar values as found with gyrokinetic simulations. Moreover, the data shows a shift of in the ion-diamagnetic direction at the edge after the transition from LOC to SOC consistent with a change in the dominant turbulence regime. The upgrade of the core CXRS system provides as well a deeper insight into the intrinsic rotation. This paper shows that the reversal of the core toroidal rotation occurs clearly after the LOC–SOC transition and concomitant with the peaking of the electron density.

Neoclassical flows in deuterium–helium plasma density pedestals

In tokamak transport barriers, the radial scale of profile variations can be comparable to a typical ion orbit width, which makes the coupling of the distribution function across flux surfaces important in the collisional dynamics. We use the radially global steady-state neoclassical code Perfect (Landreman et al 2014 Plasma Phys. Control. Fusion 56 045005) to calculate poloidal and toroidal flows, and radial fluxes, in the pedestal. In particular, we have studied the changes in these quantities as the plasma composition is changed from a deuterium bulk species with a helium impurity to a helium bulk with a deuterium impurity, under specific profile similarity assumptions. In the presence of sharp profile variations, the poloidally resolved radial fluxes are important for the total fluxes to be divergence-free, which leads to the appearance of poloidal return-flows. These flows exhibit a complex radial–poloidal structure that extends several orbit widths into the core and is sensitive to abrupt radial changes in the ion temperature gradient. We find that a sizable neoclassical toroidal angular momentum transport can arise in the radially global theory, in contrast to the local.

Total fluid pressure imbalance in the scrape-off layer of tokamak plasmas

Simulations using the fully kinetic neoclassical code XGCa (X-point included guiding- center axisymmetric) were undertaken to explore the impact of kinetic effects on scrape-off layer (SOL) physics in DIII-D H-mode plasmas. XGCa is a total-f, gyrokinetic code which self-consistently calculates the axisymmetric electrostatic potential and plasma dynamics, and includes modules for Monte Carlo neutral transport.Previously presented XGCa results showed several noteworthy features, including large variations of ion density and pressure along field lines in the SOL, experimentally relevant levels of SOL parallel ion flow (Mach number ∼ 0.5), skewed ion distributions near the sheath entrance leading to subsonic flow there, and elevated sheath potentials (Churchill 2016 Nucl. Mater. Energy 1–6).In this paper, we explore in detail the question of pressure balance in the SOL, as it was observed in the simulation that there was a large deviation from a simple total pressure balance (the sum of ion and electron static pressure plus ion inertia). It will be shown that both the contributions from the ion viscosity (driven by ion temperature anisotropy) and neutral source terms can be substantial, and should be retained in the parallel momentum equation in the SOL, but still falls short of accounting for the observed fluid pressure imbalance in the XGCa simulation results.

Dependence of the L–H transition on X-point geometry and divertor recycling on NSTX

The edge electron (Te) and ion temperature (Ti) at the time of the L–H transition increase when the X-point radius (RX) is reduced to a high-triangularity shape while maintaining constant edge density. Consequently the L–H power threshold (PLH) is larger for the high-triangularity shape. This supports the prediction that a single-particle loss hole, whose properties are strongly linked to RX and Ti, influences the edge radial electric field (Er) and Er × B flow-shearing rate available for turbulence suppression. Simulations using XGC0, a full-f drift-kinetic neoclassical code, indicate that maintaining a constant Er × B flow-shearing rate does require a larger heat flux and edge Ti as RX decreases. NSTX also observes a decrease in PLH when the divertor recycling is decreased using lithium coatings. However, the edge Te and Ti at the L–H transition appear independent of the divertor recycling for a constant shape. XGC0 calculations demonstrate that more heat flux is needed to maintain the edge Ti and the Er × B flow-shearing rate as the contribution of divertor recycling to the overall neutral fuelling rate increases.

On neoclassical impurity transport in stellarator geometry

The impurity dynamics in stellarators has become an issue of moderate concern due to the inherent tendency of the impurities to accumulate in the core when the neoclassical ambipolar radial electric field points radially inwards (ion root regime). This accumulation can lead to collapse of the plasma due to radiative losses, and thus limit high performance plasma discharges in non-axisymmetric devices.A quantitative description of the neoclassical impurity transport is complicated by the breakdown of the assumption of small E × B drift and trapping due to the electrostatic potential variation on a flux surface compared with those due to the magnetic field gradient. This work examines the impact of this potential variation on neoclassical impurity transport in the Large Helical Device heliotron. It shows that the neoclassical impurity transport can be strongly affected by . The central numerical tool used is the δf particle in cell Monte Carlo code EUTERPE. The used in the calculations is provided by the neoclassical code GSRAKE. The possibility of obtaining a more general self-consistently with EUTERPE is also addressed and a preliminary calculation is presented.

The role of the radial electric field for the transition to high confinement regimes

The radial electric field Er(x, t), and particularly its gradient, has been invoked by various theories and empirical models as a crucial parameter ‘per se’ for determining the transition to high confinement regimes, such as the onset of an internal transport barrier (ITB) in the plasma core and of the H-mode pedestal at the plasma edge. This idea, however, does not consider the basic fact that in most experiments the transition to a steady-state higher confinement regimes is produced by applying sufficient additional heating onto a given target density and current profile. In order to test this ansatz on a more routine basis, we have developed here an analytical approximation to the neoclassical calculation of the radial electric field, adapted for the 2D toroidal geometry of JET to describe all collisionality regimes (banana, banana-plateau, Pfirsch–Schlüter) and to include averaging over the potato orbits. An analytic calculation of the error bars on Er(x, t) has also been developed, which has allowed us to compare and successfully benchmark our calculations with the results of neoclassical codes such as JETTO and NCLASS. We are then able to demonstrate a striking similarity in the shape of Er(x, t) in steady-state L-mode, H-mode and ITB plasmas when normalizing Er(x, t) with respect to the total heating power flux. This clearly indicates that, experimentally, there is no direct causality relation between changes in Er(x, t) and steady-state improved confinement, as these are brought about together by changes in the power deposition profile. Only two cases do not satisfy this general rule. First, localized and rapid transients (i.e. occurring on time scales much shorter than the momentum and energy confinement time) could be linked to non-neoclassical changes in Er(x, t), possibly due to turbulence suppression mechanisms. Second, when comparing H-mode plasmas with forward and reversed ion ∇B-drift direction, we demonstrate the role of prompt fast ion losses in affecting Er(x, t), most likely due to the different edge flows and their cascading effect onto the core plasma.

Ideals and Ideas: Platonism inLa Princesse de Clèves

Religious motif, almost totally absent as it is from the narrative of Mme de Lafayette's La Princesse de Cl`ves, cannot by itself stand as the foundation of the protagonist's moral choices. This article undertakes the study of the two parallel ideological influences on the thematics of the heroine's thought. One is the evident Platonism of the Jansenists and Saint Augustine, which was current in Christian thought in seventeenth century France. The other, more pertinent one is the secular Neoplatonism of the Baroque as it derives from Plato's Dialogues. Finally, the opposition between the Princesse's doubly Platonic idealism and her suitor's Aristotelian, neoclassical code of ethics is brought to light.

Numerical determination of the ambipolar electric field in a stellarator-reactor plasma

A numerical parametric study of the radial ambipolar electric field in a stellarator reactor has been undertaken. With the numerical neoclassical code FLOCS (Flow Code for Stellarators), which is capable of handling both ions and electrons of all relevant kinetic energies, the radial ambipolar field (Er)AMB is determined from the algebraic condition that ion and electron fluxes are equal. As expected, the potential is of the same order of magnitude as the temperature. Somewhat surprisingly at first sight, however, the potential does not change much with the temperature (in the parameter range under consideration), being somewhat insensitive to moderate variations of T. An explanation for this behaviour is presented. Finally, the radial particle fluxes, consistent with the obtained (Er)AMB, and the particle confinement time are computed.