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The curvature effects on the dynamics of magnetic island evolution in tokamaks are investigated both theoretically and numerically. By taking into account perpendicular and parallel heat diffusion, a new dispersion relation is derived for tearing modes that match the linear and nonlinear results. This evolution equation allows a quantitative description over the whole range of island sizes. It predicts a nonlinear instability, i.e., growing magnetic islands in linearly stable magnetic configurations. All these predictions are in excellent agreement with full tridimensional linear and nonlinear magnetohydrodynamic (MHD) computations with the latest version of XTOR [K. Lerbinger and J. F. Luciani, J. Comput. Phys. 97, 444 (1991)]. These results have important consequences on the onset of neoclassical tearing modes because they predict a resistive MHD threshold.

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The linear resistive magnetohydrodynamical stability of the n= 1 internal kink mode in tokamaks is studied numerically. The stabilizing influence of small aspect ratio [Holmes et al., Phys. Fluids B 1, 788 (1989)] is confirmed, but it is found that shaping of the cross section influences the internal kink mode significantly. For finite pressure and small resistivity, curvature effects at the q= 1 surface make the stability sensitively dependent on shape, and ellipticity is destabilizing. Only a very restricted set of finite pressure equilibria is completely stable for q. < 1. A typical result is that the resistive kink mode is slowed down by toroidal effects to a weak resistive tearing/interchange mode. It is suggested that weak resistive instabilities are stabilized during the ramp phase of the sawteeth by effects not included in linear resistive magnetohydrodynamics. Possible mechanisms for triggering a sawtooth crash are discussed.

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Stability limits for the internal kink mode are calculated for tokamaks with different current profiles and plasma cross-sections using ideal magnetohydrodynamics (MHD). The maximum stable poloidal beta at the q = 1 surface (0,) is sensitive to the current profile, but for circular cross-sections it is typically between 0.1 and 0.2. Large aspect ratio theory gives similar predictions when the appropriate boundary conditions are applied at the plasma-vacuum surface. It is found that the internal kink is significantly destabilized by ellipticity. For JET geometry, the ideal MHD limit in 0, is typically between 0.03 and 0.1, and arbitrarily low limits can result if the shear is reduced at the q = 1 surface. A large aspect ratio expansion of the Mercier criterion retaining the effects of ellipticity and triangularity is used to illustrate the destabilizing influence of ellipticity.

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