Strong gradient regions in tokamaks such as the pedestal or internal transport barriers are regions of reduced turbulence where neoclassical transport can play a dominant role. In pedestals, gradient lengths comparable to the ion poloidal gyroradius have been measured. Standard neoclassical theory can miss important strong gradient effects in these regions because it assumes that the gradient length scales of density, temperature and potential are larger than the ion poloidal gyroradius. We extend plateau regime neoclassical theory into regions of gradients of the order of the ion poloidal gyroradius to capture strong gradient effects on transport processes in the pedestal and internal transport barriers. The fundamental idea behind our new framework is to keep a scale separation between the orbit widths and the gradient length scales by performing a large aspect ratio expansion. In the plateau regime, strong gradients cause poloidal variation that is in-out as well as up-down asymmetric. We study two different test cases assuming either radial force balance or the absence of turbulence and show that strong gradient effects can enhance or reduce standard neoclassical theory predictions in the plateau regime in strong gradient regions.

Recent nonlinear gyrokinetic simulations have shown that impurities can strongly modify the turbulent heat flux in stellarator plasmas. Here, the ion-temperature-gradient (ITG) dispersion relation in a plasma containing impurities is analytically solved in certain limits and an expression for the modification of the ITG growth rate by impurities is derived. The analytical expression is the sum of three terms corresponding to three different physical causes (impurity density gradient, impurity temperature gradient and dilution) of the change in the growth rate. The scalings predicted analytically for the modification of the growth rate are shown to be reproduced by linear gyrokinetic simulations. The conditions for reduction or increase of the ITG growth by impurities are also correctly predicted by the analytical solution to the dispersion relation. Finally, a remarkable correlation is found between the analytical expression for the modification of the growth rate and the modification of the turbulent heat flux obtained from nonlinear gyrokinetic simulations.

Until now, quasi-isodynamic magnetic fields have been the only known stellarator configurations that, at low collisionality, give small radial neoclassical transport and zero bootstrap current for arbitrary plasma profiles, the latter facilitating control of the magnetic configuration. The recently introduced notion of piecewise omnigenous fields has enormously broadened the space of stellarator configurations with small radial neoclassical transport. In this Letter, the existence of piecewise omnigenous fields that give zero bootstrap current is proven analytically and confirmed numerically. These results establish piecewise omnigenity as an alternative approach to stellarator reactor design.

Stellarator magnetic fields that are perfectly optimized for neoclassical transport (with levels of radial neoclassical transport comparable to tokamaks) are called omnigenous. Quasi-isodynamic (QI) magnetic fields are a subset of omnigenous magnetic fields in which the isolines of the magnetic field strength close poloidally, which grants them the additional property of producing zero bootstrap current. A frequent strategy in the quest for QI configurations is to optimize the magnetic field indirectly by minimizing proxies that vanish in an exactly QI field. The recently developed code MONKES enables fast computations of the neoclassical radial transport and bootstrap current monoenergetic coefficients, and therefore facilitates enormously to assess the efficiency of such indirect approach. By evaluating the large database of intermediate configurations that led to the configuration CIEMAT-QI, the inefficiency of the indirect optimization strategy for minimizing the bootstrap current is illustrated. In addition, MONKES is used to take the first steps in the exploration of a region of the configuration space of piecewise omnigenous fields, a novel family of optimized magnetic fields that has broadened the configuration space of stellarators with low levels of radial neoclassical transport.

Efficient control of turbulent heat transport is crucial for magnetic confinement fusion reactors. This work discusses the complex interplay between density gradients and micro-instabilities, shedding light on their impact on turbulent heat transport in different fusion devices. In particular, the influence of density gradients on turbulent heat transport is investigated through an extensive inter-machine study, including various stellarators such as W7-X, LHD, TJ-II and NCSX, along with the Asdex-Upgrade tokamak and the tokamak geometry of the Cyclone Base Case (CBC). Linear and nonlinear simulations are performed employing the $\delta$f-gyrokinetic code stella across a wide range of parameters to explore the effects of density gradients, temperature gradients, and kinetic electrons. A strong reduction in ion heat flux with increasing density gradients is found in NCSX and W7-X due to the stabilization of temperature-gradient-driven modes without significantly destabilizing density-gradient-driven modes. In contrast, the tokamaks exhibit an increase in ion heat flux with density gradients. Notably, the behavior of ion heat fluxes in stellarators does not align with that of linear growth rates. Additionally, this study provides physical insights into the micro-instabilities, emphasizing the dominance of trapped-electron-modes in CBC, AUG, TJ-II, LHD and NCSX, while both the trapped-electron-mode and the passing-particle-driven universal instability contribute significantly in W7-X.

Recent simulations have shown that, even when the magnetic field of a stellarator possesses nested toroidal flux surfaces, the orbits of passing energetic particles can exhibit islands. These 'drift islands' arise near rational flux surfaces, where they are likely to enhance alpha-particle transport – flattening the alpha density profile locally – unless they can be avoided by suitable design of the stellarator magnetic field. To investigate how this might be achieved, we derive an equation for the drift-island shape in a general stellarator. This result follows from the solution to a more fundamental problem: that of calculating the orbits of passing particles near a rational flux surface. We show that these orbits are determined by conservation of an adiabatic invariant associated with the closed rational-surface field lines. We use this 'transit adiabatic invariant' to prove that there are no drift islands, for all passing particles, if and only if the magnetic field satisfies a weaker version of the Cary-Shasharina condition for omnigeneity; we call such magnetic fields 'cyclometric'. The drift-island width scales as ∼ (ρ_/s)^1/2 a (ρ_⋆ is the normalized gyroradius, δ is the deviation from cyclometry, s is the magnetic shear, and a is the minor radius), so large drift islands could arise in low-shear stellarators that are insufficiently cyclometric. To ensure accurate results for very energetic particles, we compute higher-order corrections to the transit invariant. Our calculations agree extremely well with ASCOT5 guiding-centre and full-orbit simulations of alpha particles in reactor-scale equilibria, even at 3.5MeV. Finally, we show how our results can also be derived using Hamiltonian perturbation theory, which provides a systematic framework for calculating passing-particle orbits on both rational and irrational surfaces.

CIEMAT-QI4 is a quasi-isodynamic stellarator configuration that simultaneously features very good fast-ion confinement in a broad range of beta values, low neoclassical transport and bootstrap current, and ideal magnetohydrodynamic stability up to beta=5%. In this paper it is shown that CIEMAT-QI4 also exhibits reduced turbulent transport. This is demonstrated through nonlinear electrostatic simulations with the gyrokinetic code stella, including kinetic ions and electrons. The relation between reduced turbulent transport and the fact that CIEMAT-QI4 very approximately satisfies the so-called maximum-J property is discussed.

A systematic study of the impact of impurities on the turbulent heat fluxes is presented for the stellarator Wendelstein 7-X (W7-X) and, for comparison, the Large Helical Device and ITER. By means of nonlinear multispecies gyrokinetic simulations, it is shown that impurities, depending on the sign of their density gradient, can significantly enhance or reduce turbulent ion heat losses. For the relevant scenario of turbulence reduction, an optimal impurity concentration that minimizes the ion heat diffusivity emerges as a universal feature. This result demonstrates the potential of impurities for controlling turbulence and accessing enhanced confinement regimes in fusion plasmas and, in particular, in W7-X.

After a long device enhancement phase, scientific operation resumed in 2022. The main new device components are the water cooling of all plasma facing components and the new water-cooled high heat flux divertor units. Water cooling allowed for the first long-pulse operation campaign. A maximum discharge length of 8 min was achieved with a total heating energy of 1.3 GJ. Safe divertor operation was demonstrated in attached and detached mode. Stable detachment is readily achieved in some magnetic configurations but requires impurity seeding in configurations with small magnetic pitch angle within the edge islands. Progress was made in the characterization of transport mechanisms across edge magnetic islands: Measurement of the potential distribution and flow pattern reveals that the islands are associated with a strong poloidal drift, which leads to rapid convection of energy and particles from the last closed flux surface into the scrape-off layer. Using the upgraded plasma heating systems, advanced heating scenarios were developed, which provide improved energy confinement comparable to the scenario, in which the record triple product for stellarators was achieved in the previous operation campaign. However, a magnetic configuration-dependent critical heating power limit of the electron cyclotron resonance heating was observed. Exceeding the respective power limit leads to a degradation of the confinement.

The first experimental campaigns of Wendelstein 7-X (W7-X) have shown that turbulence plays a decisive role in the performance of neoclassically optimized stellarators. This stresses the importance of understanding microturbulence from the theoretical and experimental points of view. To this end, this paper addresses a comprehensive characterization of the turbulent fluctuations by means of nonlinear gyrokinetic simulations performed with the code stella in two W7-X scenarios. In the first part of the paper, the amplitude of the density fluctuations is calculated and compared with measurements obtained by Doppler reflectometry (DR) in the OP1 experimental campaigns. It is found that the trend of the fluctuations along the radius is explained by the access of the DR system to different regions of the turbulence wavenumber spectrum. In the second part of the article, frequency spectra of the density fluctuations and the zonal component of the turbulent flow are numerically characterized for comparisons against future experimental analyses. Both quantities feature broad frequency spectra with dominant frequencies of O(1)-O(10) kHz.

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/

In omnigenous magnetic fields, charged particles are perfectly confined in the absence of collisions and turbulence. For this reason, the magnetic configuration is optimized to be close to omnigenity in any candidate for a stellarator fusion reactor. However, approaching omnigenity imposes severe constraints on the spatial variation of the magnetic field. In particular, the topology of the contours of constant magnetic field strength on each magnetic surface must be such that there are no particles transitioning between different types of wells. This, in turn, usually leads to complicated plasma shapes and coils. This Letter presents a new family of optimized fields that display tokamak-like collisional energy transport while having transitioning particles. This result radically broadens the space of accessible reactor-relevant configurations.

We provide an overview of activities carried out at the TJ-II stellarator for improving our understanding of- and developing plasma physics models for particle density profiles in stellarators. Namely, we report on recent progress in turbulent particle transport simulation, validation of pellet deposition models, density profile shaping for performance control and new experimental techniques for edge turbulence and plasma-neutral interaction.

We present a new neoclassical transport model for large aspect ratio tokamaks where the gradient scale lengths are of the size of the ion poloidal gyroradius. Previous work on neoclassical transport across transport barriers assumed large density and potential gradients but a small temperature gradient, or neglected the gradient of the mean parallel flow. Using large aspect ratio and low collisionality expansions, we relax these restrictive assumptions. We define a new set of variables based on conserved quantities, which simplifies the drift kinetic equation whilst keeping strong gradients, and derive equations describing the transport of particles, parallel momentum and energy by ions in the banana regime. The poloidally varying parts of density and electric potential are included. Studying contributions from both passing and trapped particles, we show that the resulting transport is dominated by trapped particles. We find that a non-zero neoclassical particle flux requires parallel momentum input which could be provided through interaction with turbulence or impurities. We derive upper and lower bounds for the energy flux across a transport barrier in both temperature and density and present example profiles and fluxes.

A new quasi-isodynamic (QI) stellarator configuration optimized for the confinement of energetic ions at low plasma β is obtained. The numerical optimization is carried out using the STELLOPT suite of codes. New proxies to measure closeness to quasi-isodynamicity and quality of fast ion confinement have been included. The new configuration has poloidally closed contours of magnetic field strength, low magnetic shear and a rotational transform profile allowing an island divertor. It shows ideal and ballooning magnetohydrodynamic stability up to β = 5 % , reduced effective ripple, with ϵ e f f < 0.5 % in the plasma core. Even at low β , the configuration approximately satisfies the maximum- J property, and the confinement of fast ions is good at β ∼ 1.5 % and becomes excellent at reactor values, β ∼ 4 % . An evaluation of the D 31 neoclassical mono-energetic coefficient supports the expectation of a reduced bootstrap current for plasmas confined in QI configurations. A set of filamentary coils that preserve the good confinement of fast ions in the core is presented.

Due to errors in the production process, not attributable to the authors, there are three mistakes in the original text: On page 6, column 1, line 11, ‘with the standard and HM configurations of W7-X’ should be ‘with the STD and HM configurations of W7-X’. On page 9, column 2, line 10 (from the bottom), ‘for the STD of W7-X than for the HM one’ should be ‘for the STD configuration of W7-X than for the HM one’.

In reactor-relevant plasmas, neoclassical transport drives an outward particle flux in the core of large stellarators and predicts strongly hollow density profiles. However, this theoretical prediction is contradicted by experiments. In particular, in Wendelstein 7-X, the first large optimized stellarator, flat or weakly peaked density profiles are generally measured, indicating that neoclassical theory is not sufficient and that an inward contribution to the particle flux is missing in the core. In this Research Letter, it is shown that the turbulent contribution to the particle flux can explain the difference between experimental measurements and neoclassical predictions. The results of this Research Letter also prove that theoretical and numerical tools are approaching the level of maturity needed for the prediction of equilibrium density profiles in stellarator plasmas, which is a fundamental requirement for the design of operation scenarios of present devices and future reactors.

Stellarator magnetic configurations need to be optimized in order to meet all the required properties of a fusion reactor. In this work, it is shown that a flat-mirror quasi-isodynamic (QI) configuration (i.e. a QI configuration with sufficiently small radial variation of the mirror term) can achieve small radial transport of energy and good confinement of bulk and fast ions even if it is not very close to perfect omnigeneity, and for a wide range of plasma scenarios, including low β and small radial electric field. This opens the door to constructing better stellarator reactors. On the one hand, they would be easier to design, as they would be robust against error fields. On the other hand, they would be easier to operate since, both during startup and steady-state operation, they would require less auxiliary power, and the heat loads on plasma-facing components caused by fast ion losses would be reduced to acceptable levels.

The initial operation of neutral beams in Wendelstein 7-X enabled the first measurements of carbon (C 6+ ) temperature and rotation with a multi-chord charge exchange recombination spectroscopy system. In this article we analyse the velocity measurements obtained with this system to infer the profiles of radial electric field, E r , and net parallel velocity of fully stripped carbon ions, V ‖ . We include a general treatment of the flow geometry and the velocity corrections due to the energy dependence of the charge exchange cross section. The inverted profiles are compared with neoclassical calculations for various plasma conditions and two magnetic configurations. It is found that both the E r and V ‖ experimental profiles agree in sign and order of magnitude with the neoclassical expectations and reproduce the expected dependencies on plasma parameters and magnetic configuration, within the reduced dataset presented. However, quantitative differences with respect to the neoclassical profiles are observed above the estimated errors in various conditions.