Unified theory of the semi-collisional tearing mode and internal kink mode in a hot tokamak: implications for sawtooth modelling

In large hot tokamaks like JET, the width of the reconnecting layer for resistive modes is determined by semi-collisional electron dynamics and is much less than the ion Larmor radius. Firstly a dispersion relation valid in this regime is derived which provides a unified description of drift-tearing modes, kinetic Alfv?n waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilizing ion orbit effect, obtained previously by Cowley et al (1986 Phys. Fluids 29 3230), which implies large values of the tearing mode stability parameter ?? are required for instability. Secondly, at high beta it is shown that the tearing mode interacts with the kinetic Alfv?n wave and that there is an absolute stabilization for all ?? due to the shielding effects of the electron temperature gradients, extending the result of Drake et al (1983 Phys. Fluids 26 2509) to large ion orbits. The nature of the transition between these two limits at finite values of beta is then elucidated. The low beta formalism is also relevant to the m?=?n?=?1 tearing mode and the dissipative internal kink mode, thus extending the work of Pegoraro et al (1989 Phys. Fluids B 1 364) to a more realistic electron model incorporating temperature perturbations, but then the smallness of the dissipative internal kink mode frequency is exploited to obtain a new dispersion relation valid at arbitrary beta. A diagram describing the stability of both the tearing mode and dissipative internal kink mode, in the space of ?? and beta, is obtained. The trajectory of the evolution of the current profile during a sawtooth period can be plotted in this diagram, providing a model for the triggering of a sawtooth crash.