On the confinement of a tokamak plasma

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The goal of this paper is to understand from a mathematical point of view the magnetic confinement of plasmas for fusion. Following Frénod and Sonnendrücker FS2, we first use two-scale convergence tools to derive a gyrokinetic system for a plasma submitted to a large magnetic field with a slowly spatially varying intensity. We formally derive from this system a simplified bi-temperature fluid system. We then investigate the behaviour of the plasma in such a regime and we prove nonlinear stability or instability depending on which side of the tokamak we are looking at. In our analysis, we will also point out that there exists a temperature gradient threshold beyond which one can expect stability, even in the "bad" side: this corresponds to the so-called H-mode.