The theory of the early nonlinear stage of reconnection, when the plasma central core displacement ξ0 is still small with respect to the q=1 radius r1 but exceeds the characteristic width of the singular layer, is considered. The two-dimensional magnetic geometry of m=1 reconnection is determined by solving Waelbroeck’s equations [Phys. Fluids B 1, 2372 (1989)]. Fast nonlinear exponential growth of the mode in contemporary high-temperature experiments is explained on the basis of the nonlinear two-fluid magnetohydrodynamic theory. It is shown that the stabilization of the reconnecting m=1 mode by ω*-effects is in qualitative agreement with experimental observations. A nonlinear cancellation of the ω*-stabilization is proposed as explanation of the triggering of the sawtooth crash.

Oscillation bursts (fishbones) of magnetically confined plasmas are associated with the excitation of an m0=1 mode when the ideal magnetohydrodynamic (MHD) threshold for the instability of this mode is reached. Near this threshold and in the absence of an effective ‘‘viscous’’ dissipative process, this mode remains marginally stable as a result of finite ion Larmor radius effects and has a real frequency of oscillation near the ion diamagnetic frequency. The release of the mode excitation energy related to the gradient of the plasma pressure, in the case of perpendicular neutral beam injection, is allowed by the resonant interaction of the mode with fast trapped ions that precess around the torus as a result of the curvature and the gradient of the confining magnetic field. This process plays the role of an effective viscosity. A consequence of the presented interpretation is that fishbone oscillations may also be excited in the case of parallel beam injection. In fact, for relatively low values of the beam transverse pressure, the basic mode frequency is related to the parameters of the target plasma and does not depend on the magnetic drift frequency of the beam particles that are injected nor on their velocity distribution.

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Fusion-produced alpha particles may spontaneously enhance the stability of an ignited plasma against m=1 internal modes. Stable values of the poloidal beta of the thermal plasma component significantly in excess of the ideal magnetohydrodynamic threshold can be attained, provided the region where the magnetic helical parameter q is below unity is not too wide and the ignition temperature is not too high. A comprehensive analysis of the different instability regimes is presented, with special attention to so-called ‘‘fishbone oscillations’’ and their influence on the ignition energy balance. The theoretical predictions compare favorably with sawtooth suppression experiments in the JET tokamak [Phys. Rev. Lett. 60, 2148 (1988)] with high-power, ion-cyclotron resonant frequency heating, where energetic ions in the MeV range are produced.

A review of the present theoretical understanding of the linear stability of internal m=1 modes is presented and its connection to phenomena observed in toroidal magnetic confinement experiments, i.e. 'sawtooth' and 'fishbone' oscillations, is discussed. Particular attention is devoted to the analysis of non-magnetohydrodynamic (non-MHD) effects, such as those due to finite diamagnetic and electron drift frequencies and to finite ion Larmor radius, and to the special role played by energetic particles (whose response is primarily kinetic and which can stabilize low frequency modes and destabilize new, higher frequency, branches). Some recent developments in the non-linear theory of these modes are also discussed. This review complements that of Kuvshinov and Savrukhin (Sov. J. Plasma Phys. 16 (1990) 353)