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Publication# First global simulations of plasma turbulence in a stellarator with an island divertor

Résumé

We present the results of 3D, flux-driven, global, two-fluid electrostatic turbulence simulations in a 5-field period stellarator with an island divertor. The numerical simulations are carried out with the GBS code, which solves the two-fluid drift-reduced Braginskii equations and has been extended recently to simulate plasma turbulence in non-axisymmetric magnetic equilibria. The vacuum magnetic field used in the simulations is carefully constructed using Dommaschk potentials in order to describe a configuration with a central region of nested flux surfaces, surrounded by a chain of magnetic islands. In a similar way to the diverted configurations of W7-AS and W7-X, particles and heat, transported radially outwards from the core region, reach the island region, which effectively acts as a scrape-off-layer with the open field lines striking the walls at specific toroidal locations of the device wall. We find that the radial particle and heat transport is mainly driven by a field-aligned mode with low poloidal wavenumber, whose origin is investigated theoretically. The equilibrium radial electric field in the core is found to be in the ion-root regime, Er

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António João Caeiro Heitor Coelho, Maurizio Giacomin, Joaquim Loizu Cisquella, Paolo Ricci

Results of a three-dimensional, flux-driven, electrostatic, global, two-fluid turbulence simulation for a five-field period stellarator with an island divertor are presented. The numerical simulation is carried out with the GBS code, recently extended to simulate plasma turbulence in non-axisymmetric magnetic equilibria. The vacuum magnetic field used in the simulation is generated with the theory of Dommaschk potentials, and describes a configuration with a central region of nested flux surfaces, surrounded by a chain of magnetic islands, similar to the diverted configurations of W7-X. The heat outflowing from the core reaches the island region and is transported along the magnetic islands, striking the vessel walls, which correspond to the boundary of the simulation domain. The radial transport of particles and heat is found to be mainly driven by a field-aligned coherent mode with poloidal number m = 4. The analysis of this mode, based on non-local linear theory considerations, shows its ballooning nature. In contrast to tokamak simulations and experiments, where blobs often contribute to transport, we do not observe the presence of intermittent transport events.

2022The goal of thermonuclear fusion research is to provide power plants, that will be able to produce one gigawatt of electricity. Among the different ways to achieve fusion, the tokamak, based on magnetic confinement, is the most promising one. A gas is heated up to hundreds of millions of degrees and becomes a plasma, which is maintained – or confined – in a toroidal vessel by helical magnetic field lines. Then, deuterium and tritium are injected and fuse to create an α particle and an energetic neutron. In order to have a favorable power balance, the power produced by fusion reactions must exceed the power needed to heat the plasma and the power losses. This can be cast in a very simple expression which stipulates that the product of the density, the temperature and the energy confinement time must exceed some given value. Unfortunately, present-days tokamaks are not able to reach this condition, mostly due to plasma turbulence. The latter phenomenon enhances the heat losses and degrades the energy confinement time, which cannot be predicted by analytical theories such as the so-called neoclassical theory in which the heat losses are caused by Coulomb collisions. Therefore, numerical simulations are being developed to model plasma turbulence, mainly caused by the Ion and Electron Temperature-Gradient and the Trapped-Electron-Mode instabilities. The plasma is described by a distribution function which evolves according to the Vlasov equation. The electromagnetic fields created by the particles are self-consistently obtained through Maxwell's equations. The resulting Vlasov-Maxwell system is greatly simplified by using the gyrokinetic theory, which consists, through an appropriate ordering, of eliminating the fast gyromotion (compared to the typical frequency of instabilities). Nevertheless, it is still extremely difficult to solve this system numerically due to the large range of time and spatial scales to be resolved. In this thesis, the Vlasov-Maxwell system is solved in the electrostatic and collisionless limit with the Particle-In-Cell (PIC) ORB5 code in global tokamak geometry. This Monte-Carlo approach suffers from statistical noise which unavoidably degrades the quality of the simulation. Consequently, the first part of this work has been devoted to the optimization of the code with a view to reduce the numerical noise. The code has been rewritten in a new coordinate system which takes advantage of the anisotropy of turbulence, which is mostly aligned with the magnetic field lines. The overall result of the optimization is that for a given accuracy, the CPU time has been decreased by a factor two thousand, the total memory has been decreased by a factor ten and the numerical noise has been reduced by a factor two hundred. In addition, the scaling of the code with respect to plasma size is presently optimal, suggesting that ORB5 could compute heat transport for future fusion devices such as ITER. The second part of this thesis presents the validation of the code with numerical convergence tests, linear (including dispersion relations) and nonlinear benchmarks. Furthermore, the code has been applied to important issues in gyrokinetic theory. It is shown for the first time that a 5D global delta-f PIC code can achieve a thermodynamic steady state on the condition that some dissipation is present. This is a fundamental result as the main criticism against delta-f PIC codes is their inability to deal with long time simulations. Next, the role of the parallel nonlinearity is studied and it is demonstrated in this work that this term has no real influence on turbulence, provided the numerical noise is sufficiently low. This result should put an end to the controversy that recently occurred, in which gyrokinetic simulations using different numerical approaches yielded contradictory results. Finally, thanks to the optimization of the code, the gyrokinetic model has been extended to include the kinetic response of trapped-electrons, in place to the usual adiabatic (Boltzmann) approximation. For the first time, global TEM nonlinear simulations are presented, and the role of the zonal flow on heat transport is analyzed. This study will help in acquiring some knowledge on the less-known TEM turbulence (as compared to ITG). In conclusion, this thesis is one of the main steps of the development of ORB5, which is now a state-of-the-art gyrokinetic code for collisionless ITG and TEM turbulence, and has brought several contributions to the understanding of these phenomena.

In magnetic fusion devices, the radial transport of heat and particle largely exceeds predictions based on collisional processes. This is widely understood as a consequence of small-scale turbulence which results from the nonlinear behaviour of so-called microinstabilities. The complexity of such nonlinear phenomena allows one to address microturbulence only with a numerical description, carried out here within the gyrokinetic framework. This reduced kinetic model describes the evolution of the particle distribution functions and of the self-consistently generated electromagnetic fields neglecting the fast gyromotion. In this work we applied the grid-based gyrokinetic code GENE, using both its local and global versions, to model some of the experimental observations made in the Tokamak à Configuration Variable (TCV) at the Swiss Plasma Center. All simulations are performed considering realistic magnetic geometries, in turn provided by the MHD equilibrium solver CHEASE. In order to verify the interface of GENE with CHEASE, a series of benchmarks have been developed and successfully carried out in the linear local limit. These tests have then been extended to the global version of the code and a good agreement found with results obtained with the gyrokinetic Particle In Cell code ORB5. A significant part of this work deals with the electron heat confinement improvement observed when the shape of the plasma is modified by changing the sign of the edge triangularity, from positive to negative. In the latter case, half the heating power is required to maintain the same electron profiles compared to the former, which was experimentally interpreted as a better confinement at all radial locations, even though triangularity has a finite radial penetration depth. A series of local runs were carried out to investigate the dependence of profile stiffness on shaping, failing at reproducing both the absolute level of transport as well as the ratio between the two shapes. Global gradient-driven simulations have then been performed, showing a very high sensitivity of the electron heat flux with respect to the density gradient. These runs, carried out neglecting carbon impurities, are compatible with the experiments when using parameters from an experimentally well diagnosed discharge. In this case, strong global effects which lower the heat flux compared to local runs are seen. Local and global simulations have then been performed looking at axisymmetric dynamics in the frequency range of the Geodesic Acoustic Mode (GAM). Experimentally, the GAM is almost always observed as a radially coherent mode, i.e. an oscillation at constant frequency over a main fraction of the plasma minor radius. The only exception is for very large values of the edge safety factor q, where the mode looses its coherence becoming dispersive. A density ramp-up was studied with local simulation, already obtaining a reasonable agreement with measurements of heat transport as well as GAM frequency and amplitude. The coherent GAM was then investigated with global runs. Simulations qualitatively agree with experiments and a good match is recovered with ORB5 results when the same physical model is used. Finally, the hypothesis of a coherent-dispersive GAM transition related to the safety factor profile is addressed. It is found that changing only q is not sufficient to induce a regime transition, which thus appears to be due to other parameters, including finite machine size effects.

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