J. H. E. Proll

Small-scale turbulence originating from microinstabilities limits the energy confinement time in magnetic confinement fusion. Here we develop a semi-analytical dispersion relation based on lowest-order solutions to the gyrokinetic equations in an asymptotic expansion in the ratio of transit (bounce) frequency to the mode frequency for ions (electrons), capable of describing two common instabilities: the ion temperature gradient (ITG) mode and trapped-electron mode (TEM), in the electrostatic limit. The dispersion relation, which is valid in arbitrary toroidal geometry, takes into account resonances with the magnetic ion and bounce-averaged electron drifts, incorporates non-local effects along the magnetic field line, is valid for arbitrary sign of the growth rate and magnetic curvature, and is shown to satisfy a variational property. Several common approximation models are introduced for both the magnetic drift and finite Larmor radius (FLR) damping, with the Padé approximation for FLR effect in particular resulting in remarkable agreement with the baseline dispersion relation model at significantly reduced costs. The baseline model is verified by comparing solutions of the dispersion relation model to high-fidelity linear gyrokinetic simulations, where the exact eigenfunction of the electrostatic potential from simulations is used as a trial function, showing good quantitative agreement for ITGs and TEMs in (shaped) tokamaks as well as low-magnetic-shear stellarators.

With the advent of neoclassically optimised stellarators, optimising stellarators for turbulent transport is an important next step. The reduction of ion-temperature-gradient-driven turbulence has been achieved via shaping of the magnetic field, and the reduction of trapped-electron mode (TEM) turbulence is adressed in the present paper. Recent analytical and numerical findings suggest TEMs are stabilised when a large fraction of trapped particles experiences favourable bounce-averaged curvature. This is the case for example in Wendelstein 7-X [C.D. Beidler $\textit{et al}$ Fusion Technology $\bf{17}$, 148 (1990)] and other Helias-type stellarators. Using this knowledge, a proxy function was designed to estimate the TEM dynamics, allowing optimal configurations for TEM stability to be determined with the STELLOPT [D.A. Spong $\textit{et al}$ Nucl. Fusion $\bf{41}$, 711 (2001)] code without extensive turbulence simulations. A first proof-of-principle optimised equilibrium stemming from the TEM-dominated stellarator experiment HSX [F.S.B. Anderson $\textit{et al}$, Fusion Technol. $\bf{27}$, 273 (1995)] is presented for which a reduction of the linear growth rates is achieved over a broad range of the operational parameter space. As an important consequence of this property, the turbulent heat flux levels are reduced compared with the initial configuration.

Microinstabilities exhibit a rich variety of behavior in stellarators due to the many degrees of freedom in the magnetic geometry. It has recently been found that certain stellarators (quasi-isodynamic ones with maximum-$J$ geometry) are partly resilient to trapped-particle instabilities, because fast-bouncing particles tend to extract energy from these modes near marginal stability. In reality, stellarators are never perfectly quasi-isodynamic, and the question thus arises whether they still benefit from enhanced stability. Here the stability properties of Wendelstein 7-X and a more quasi-isodynamic configuration, QIPC, are investigated numerically and compared with the National Compact Stellarator Experiment (NCSX) and the DIII-D tokamak. In gyrokinetic simulations, performed with the gyrokinetic code GENE in the electrostatic and collisionless approximation, ion-temperature-gradient modes, trapped-electron modes and mixed-type instabilities are studied. Wendelstein 7-X and QIPC exhibit significantly reduced growth rates for all simulations that include kinetic electrons, and the latter are indeed found to be stabilizing in the energy budget. These results suggest that imperfectly optimized stellarators can retain most of the stabilizing properties predicted for perfect maximum-$J$ configurations.