Local and Global Eulerian Gyrokinetic Simulations of Microturbulence in Realistic Geometry with Applications to the TCV Tokamak

In magnetically confined fusion devices, the energy and particle transport is significantly larger than expected from purely collisional processes. This degraded confinement mostly results from small-scale turbulence and prevents from reaching self-sustained burning plasma conditions in present day experiments. A better understanding of these nonlinear phenomena is therefore of key importance on the way towards controlled fusion. The small-scale microinstabilities and associated turbulence are investigated for Tokamak plasmas by means of numerical simulations in the frame of the gyrokinetic theory. This model describes the evolution of the particle distribution functions in phase space together with self-consistent electromagnetic fields, while neglecting the fast motion associated with the Larmor orbit of particles around the magnetic field lines. In the course of this thesis work, substantial modifications to the existing Eulerian gyrokinetic code GENE have been carried out in collaboration with the Max-Planck-Institute für Plasmaphysik in Garching, Germany. The code has been extended from a local approximation, which only considers a reduced volume of a fusion plasma, to a global version which fully includes radial temperature and density profiles as well as radial magnetic equilibrium variations. To this end, the gyrokinetic equations have been formulated for general magnetic geometry, keeping radial variations of equilibrium quantities, and considering field aligned coordinates, suitable for their numerical implementation. The numerical treatment of the radial direction has been modified from a Fourier representation in the local approach to real space in the global code. This has in particular required to adapt the radial derivatives, the field solver, and to implement a real space dealiasing scheme for the treatment of the nonlinearity. A heat source was in addition introduced to allow for steady state global nonlinear simulations. An important part of this work also focused on the description of the magnetic equilibrium. A circular concentric flux surface model as well as an interface with an MHD equilibrium code were implemented. A detailed investigation concerning the s – α model, previously used in local codes, was also carried out. It was shown that inconsistencies in this model had resulted in misinterpreted agreement between local and global results at large ρ* = ρs/a values, with ρs the Larmor radius and a the minor radius of the Tokamak. True convergence between local and global simulations was finally obtained by correct treatment of the geometry in both cases and considering the appropriate ρ* → 0 limit in the latter case. The new global code was furthermore successfully tested and benchmarked against various other codes in the adiabatic electron limit in both the linear and nonlinear regime. A nonlinear ρ* scan was in addition carried out showing convergence to the local results in the limit ρ* → 0 and also providing further insight on previous disagreements between two other global gyrokinetic codes concerning ρ* convergence. Linear global simulations with kinetic electrons have shown consistent behavior with respect to local results. Using the interface with the MHD equilibrium code, the effects of plasma shaping on Ion Temperature Gradient (ITG) instabilities were investigated by means of local simulations. A favorable influence of elongation and negative triangularity was observed. It was shown that these effects could be mostly accounted for by the modifications of the effective flux-surface averaged temperature gradient. Most importantly, a unique effective nonlinear critical temperature gradient could be determined for the different considered elongations and triangularities. The local code was finally used to investigate particle and energy transport in the case of TCV discharges presenting an electron Internal Transport Barrier (eITB). It was shown that at the transition between ITG and Trapped Electron Mode (TEM) dominated turbulent regimes, the particle flux goes to zero. Interestingly, this effect could be well reproduced by a quasi-linear approach where all the different unstable wavenumbers are considered. The nonlinear simulations also revealed that a minimum of the electron heat diffusivity is observed at the transition between the TEM and ITG regimes. A strong dependence of this quantity was also noticed with respect to the density gradient. Quantitative comparisons with experimental results have shown that a reasonable agreement could only be reached in regions where the density gradient is small while the flux tube simulations seem to overestimate the heat transport if one accounts for gradient values in the center of the transport barrier. Some first nonlinear global simulations appear to indicate that finite ρ* effects could potentially play an important role and thus reduce the heat diffusivity to realistic values.

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