On The Confinement of a Tokamak Plasma

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 [SIAM J. Math. Anal., 32 (2001), pp. 1227–1247], 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 bitemperature fluid system. We then investigate the behavior 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.