An overview is given of physics differences between stellarators and tokamaks, including magnetohydrodynamic equilibrium, stability, fast-ion physics, plasma rotation, neoclassical and turbulent transport, and edge physics. Regarding microinstabilities, it is shown that the ordinary, collisionless trapped-electron mode is stable in large parts of parameter space in stellarators that have been designed so that the parallel adiabatic invariant decreases with radius. Also, the first global, electromagnetic, gyrokinetic stability calculations done for Wendelstein 7-X suggest that kinetic ballooning modes are more stable than in a typical tokamak.

Turbulence is widely expected to limit the confinement, and thus the overall performance, of modern neoclassically-optimized stellarators. We employ novel petaflop-scale gyrokinetic simulations to predict the distribution of turbulence fluctuations and the related transport scaling on entire stellarator magnetic surfaces, and reveal striking differences to tokamaks. Using a stochastic global-search optimization method, we derive the first turbulence-optimized stellarator configuration stemming from an existing quasi-omnigenous design.Throughout the history of magnetic fusion, a recurrent theme has been the surprising sensitivity of plasma performance to the details of the magnetic field. For instance, it has long been known that the confinement of alpha particles can be spoiled by small ripples in the magnetic field. More recently, magnetic perturbations have been found to dramatically influence instabilities of the plasma edge [1]. In both stellarators and tokamaks, experiments show that the level of turbulence may be reduced by modifying the magnetic field. As notable examples, the confinement time in the TCV tokamak is doubled by reversing the triangularity of the poloidal cross section of the flux surfaces [2], and in the LHD stellarator the turbulent transport can be reduced significantly by adjusting the coil currents so as to shrink the circumference of the torus by pushing it radially inwards [3].Stellarators typically possess 40-50 degrees of freedom in the shaping of the magnetic field, an order of magnitude more than for tokamaks [4]. This enhanced flexibility can be used to optimize various plasma properties [5], and the latest demonstration of the power of such optimization is expected to be realized in the superconducting stellarator experiment Wendelstein 7-X (W7-X), in Greifswald, Germany [6]. A tantalizing possibility for fusion researchers is to try to exploit any leeway in the magnetic geometry to design configurations with better confinement. In W7-X, this has already been done for the collisional (so-called "neoclassical") transport, but no device built so far is optimized with respect to turbulence.In order to understand how energy transport depends on the magnetic-field geometry, it is helpful to numerically simulate the turbulence in a large portion of the plasma. In tokamaks, thanks to axisymmetry, restricting the computational domain to a "flux tube", a slender volume encompassing a magnetic-field line [7], suffices to calculate the transport at a radial location. In a stellarator, however, different flux tubes on a magnetic surface are not geometrically equivalent, thus it appears necessary to simulate the entire magnetic surface. Much has been learned from the flux-tube approach which, except for inspiring efforts [8], has characterized stellarator turbulence simulations to date [9][10][11][12] . Nevertheless, all these simulations have a major inherent drawback: the transport cannot be reliably determined, as the turbulence strength generally varies between different flux tubes on th...

It is shown that in perfectly quasi-isodynamic stellarators, trapped particles with a bounce frequency much higher than the frequency of the instability are stabilizing in the electrostatic and collisionless limit. The collisionless trapped-particle instability is therefore stable as well as the ordinary electron-density-gradient-driven trapped-electron mode. This result follows from the energy balance of electrostatic instabilities and is thus independent of all other details of the magnetic geometry.

This is the first of two papers about collisionless, electrostatic micro-instabilities in stellarators, with an emphasis on trapped-particle modes. It is found that, in so-called maximum-J configurations, trapped-particle instabilities are absent in large regions of parameter space. Quasi-isodynamic stellarators have this property (approximately), and the theory predicts that trapped electrons are stabilizing to all eigenmodes with frequencies below the electron bounce frequency. The physical reason is that the bounce-averaged curvature is favorable for all orbits, and that trapped electrons precess in the direction opposite to that in which drift waves propagate, thus precluding wave-particle resonance. These considerations only depend on the electrostatic energy balance, and are independent of all geometric properties of the magnetic field other than the maximum-J condition. However, if the aspect ratio is large and the instability phase velocity differs greatly from the electron and ion thermal speeds, it is possible to derive a variational form for the frequency showing that stability prevails in a yet larger part of parameter space than what follows from the energy argument. Collisionless trapped-electron modes should therefore be more stable in quasi-isodynamic stellarators than in tokamaks.

Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmed at higher density gradients by the long-wavelength coherent structures.