Spatial distribution of zonal flow drive: from tokamaks via novel hybrids to stellarators

Zonal flows are of central importance for the confinement and therefore performance of future fusion reactors that are based on the idea of toroidal magnetic confinement. By taking energy out of turbulent fluctuations and by shearing turbulent eddies apart, these flows reduce the radial turbulent transport which is responsible for the majority of the heat and particle losses in tokamaks and optimised stellarators.

Turbulent transport properties are known to be strongly affected by plasma shaping. Yet not all origins are identified, and one key missing gap in our understanding is the spatial distribution of the nonlinear drive of zonal flows due to Reynolds stress. In this work we show that this spatial distribution is generally non-trivial and generally not correlated with the distribution of turbulent fluctuations, for both axisymmetric and non-axisymmetric shaping of the nested toroidal surfaces. In particular, it is found that the zonal flow drive peaks away from the outboard midplane, where turbulent fluctuations are usually the strongest.

In pursuit of combining the advantages of tokamaks and stellarators and to bridge the gap between them, a new method was developed to optimise three-dimensionally perturbed tokamak equilibria for quasi-axisymmetry, strong enough for key stabilising benefits of stellarators to be expected. We examined the general turbulent properties in this continuous parameter space from tokamaks to stellarators, and showed that the zonal flow drive envelope here also is non-trivial, developing additional local maxima due to the localised three-dimensional shaping. Further, by extending the analysis to more traditional stellarators, the zonal flow drive is also shown to be non-trivial in the most unstable flux tube of the largest stellarator experiment, Wendelstein 7-X.

Finally, we derive and discuss how the Reynolds stress analysis can be generalised to the gyrokinetic nonlinearity, describing the zonal flow drive with kinetic effects.