A summary is given of the linear theory of the ideal and resistive hydromagnetic stability of tokamaks. The first section provides an introductory account of the various aspects of the stability problem, and the subsequent sections provide a survey of the subject and a review of the literature. For aperfectly conducting plasma the modes of instability are of three types: kink, internal, and axisymmetric. When resistivity is introduced the kink and internal modes have significantly modified forms. The analysis of the standard tokamak, having a large aspect ratio, circular cross-section and low β, is almost complete but the study of small aspect ratio, high-β configurations and the optimization of such configurations are at an early stage.

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Several tokamak experiments have exhibited an unstable behavior having the form of relaxation oscillations in which the soft-x-ray emission has a saw-tooth time dependence 1 ' 2 ' 3 . In the central region the results indicate a slowly rising temperature followed by a rapid fall. The whole process occurs repeatedly with a period of the order of a millisecond.The following general explanation of this behavior has gained some acceptance. The inner temperature rises due to Ohmic heating. The resulting increase in conductivity leads to an increase in the current density on axis and as a consequence the safety factor q falls below unity. The plasma then undergoes an instability which transports the energy which has been produced by Ohmic heating out to larger radii. In some way the plasma relaxes back to an axisymmetric state having q > 1 and the whole process is then repeated.The question arises as to precisely how this phenomenon occurs. A model has been suggested by Kadomtsev, 4 the basic elements of which are as follows, When the value of q falls below unity an m = 1 instability occurs. As a result the plasma surfaces are displaced to one side and resistivity allows a magnetic island to form on the opposite side of the plasma around the q = 1 surface. This island grows and displaces the original set of magnetic surfaces which then decay away. The resulting value of q is greater than unity but the concentration of the current toward the magnetic axis leads to a lowering of q until instability reap-12 A. Van Deursen, A. Van Lumig, and J. Reuss, Int. J. Mass. Spectrom. Ion Phys 0 18, 129 (1975).pears and the whole cycle is repeated.Our purpose here is to describe the results of a three-dimensional, nonlinear calculation which reproduces relaxation instabilities of the type observed experimentally and which demonstrates the basic features of Kadomtsev's model. The configuration studied is cylindrical and, in order to achieve acceptable computation times, a higher value of /3 has been used than is obtained in tokamak experiments.The equations solved are the time-dependent hydromagnetic equations including resistivity, viscosity, Ohmic heating, and an energy loss. The resistivity is taken to be proportional to T e " 3/2 . The viscosity used is small and is taken to be constant. The Ohm's law is E + vXB = r?(T)"j, and the resulting Ohmic heating is included together with an energy-loss term in the energy equationThe form of the energy-loss term is arbitrary but it represents an attempt to describe the more rapid energy-loss rate in the outer region.The calculations were carried out on a rectangular grid using a generalized form of the Lax-Wendroff method. Relaxation oscillations were first demonstrated on a 10x10x7 grid. The calculations described here were carried out on a more refined 14x14x10 grid and showed the A three-dimensional, nonlinear, numerical simulation is described in which saw-tooth oscillations occur as a result of a hydromagnetic relaxation instability. The time evolution of the magnetic topology demons...

Numerical simulations of tokamak plasmas are presented which reproduce the basic observed behavior of disruptions. Details are given of the onset of the disruption-initiated in the simulations by the m = 2 island intersecting the limiter or cold-gas region -and of the mechanism of the final (total or partial) plasma loss.