• magnetic confinement
• magnetic islands
• neoclassical theory
• gyrokinetic theory
• numerical simulations

Tearing modes play a key role in magnetically confined plasmas for thermonuclear interest. The nonlinear growth of these modes leads to the formation of the so-called magnetic islands. The study of magnetic islands is an important issue because they cause an increased radial heat and particle transport, thus compromising confinement, and they may also lead to major disruptions. In this thesis we addressed the study of magnetic island propagation and evolution in tokamaks, the main focus being on the drift-tearing mode. We worked on two different issues, namely the analytical study of the propagation velocity of a freely-rotating island and the self-consistent field profiles, and the numerical investigation of the regime of magnetic island propagation named "limit cycles", which has been recently observed in the Frascati tokamak FTU.

On the first subject, we managed to include the neoclassical effects in a four-field system of gyrofluid equations. To do that, we first computed the lowest-order Finite Larmor Radius (FLR) corrections to the poloidal flow damping by solving the gyrofluid equation for the ions under particular hypotheses and by using a procedure originally developed for the drift-kinetic equation. Then we used a simplified version of the gyrofluid equations originally deduced by P. B. Snyder in his PhD thesis and we closed them by including the neoclassical effects mentioned above. This system of equations was then subject to further simplifications to find the radial profiles of the relevant plasma quantities in the presence of a magnetic island. Finally, we imposed the torque-balance condition on the freely-rotating magnetic island to compute the self-consistent rotation velocity. The system of four-field gyrofluid equations we have deduced generalizes the four-field model adopted by Fitzpatrick & Waelbroeck in a series of dedicated papers and, in particular, the inclusion of the lowest-order FLR corrections to the poloidal flow damping leads to an additional term in the intermediate-damping case (which is relevant for realistic tokamak applications). The numerical resolution of the system of equations thus obtained, in principle, provides the island rotation velocity for different values of the plasma parameters entering the equations (within the limits of validity of the model).

On the second subject, we investigated the nonlinear evolution of tearing modes in the presence of a large diamagnetic velocity to reproduce the dynamics observed in the regime of island propagation named "limit cycles”. The phenomenon consists in a quasi-periodic modulation of the amplitude and the rotation frequency of magnetic islands. This regime was named "limit cycles" because of the figure drawn by the trajectory of a point in the frequency-amplitude plane. To perform this task, we numerically integrated a simplified four-field system of equations by using a finite-difference slab code developed by Daniela Grasso et al. from ISC-CNR and Politecnico di Torino. To study this particular regime of island dynamics, we performed a series of simulations in the regime of large diamagnetic velocity, corresponding to a large equilibrium density gradient. We chose to try different values of the tearing mode stability parameter Δ' and of the plasma ß, which enters our expression for the diamagnetic velocity. For every choice of Δ' we observed a reversal of the amplitude growth in the nonlinear regime, for ß large enough. For particular values of these parameters, we managed to reproduce qualitatively the amplitude evolution of the islands in the “limit cycles” regime. We also observed a significant deformation of the island and a large component of the second harmonic, in agreement with the experimental observations. We ascribed this set of peculiar features to a stabilizing effect of the diamagnetic velocity in the nonlinear regime, caused by the flattening of the density profile inside the separatrix, which is also responsible for the island deformation.