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 Alfvén waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilising ion orbit effect, obtained previously by Cowley et al. (Phys. Fluids 29 3230 1986), 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 Alfvén wave and that there is an absolute stabilisation for all ∆ ′ due to the shielding effects of the electron temperature gradients, extending the result of Drake et. al (Phys. Fluids 26 2509 1983) 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. (Phys. Fluids B 1 364 1989) 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.

The separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ′ calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ′) required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed.

It is shown that in perfectly quasi-isodynamic stellarators, trapped particles with a bounce frequency much higher than the frequency of the instability are stabilizing in the electrostatic and collisionless limit. The collisionless trapped-particle instability is therefore stable as well as the ordinary electron-density-gradient-driven trapped-electron mode. This result follows from the energy balance of electrostatic instabilities and is thus independent of all other details of the magnetic geometry.