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A new formalism for analyzing the magnetohydrodynamic stability of a limiter tokamak edge plasma is developed. Two radially localized, high toroidal mode number n instabilities are studied in detail: a peeling mode and an edge ballooning mode. The peeling mode, driven by edge current density and stabilized by edge pressure gradient, has features which are consistent with several properties of tokamak behavior in the high confinement “H”-mode of operation, and edge localized modes (or ELMs) in particular. The edge ballooning mode, driven by the pressure gradient, is identified; this penetrates ∼n1/3 rational surfaces into the plasma (rather than ∼n1/2, expected from conventional ballooning mode theory). Furthermore, there exists a coupling between these two modes and this coupling provides a picture of the ELM cycle.

A kinetic theory for magnetic islands in a low collision frequency tokamak plasma is presented. Self-consistent equations for the islands’ width, w, and propagation frequency, ω, are derived. These include contributions from the perturbed bootstrap current and the toroidally enhanced ion polarization drift. The bootstrap current is independent of the island propagation frequency and provides a drive for the island in tokamak plasmas when the pressure decreases with an increasing safety factor. The polarization drift is frequency dependent, and therefore its effect on the island stability cannot be deduced unless ω is known. This frequency is determined by the dominant dissipation mechanism, which for low effective collision frequency, νeff=ν/ε<ω, is governed by the electrons close to the trapped/passing boundary. The islands are found to propagate in the electron diamagnetic direction in which case the polarization drift is stabilizing and results in a threshold width for island growth, which is of the order of the ion banana width. At larger island widths the polarization current term becomes small and the island evolution is determined by the bootstrap current drive and Δ′ alone, where Δ′ is a measure of the magnetic free energy.

The non-relativistic theory of a plasma in an electric field E predicts that there will always be runaway electrons, although their number will be exponentially small for fields less than the Dreicer field ED. However, when E/ED ∼ kT/mec2, the ratio of the electron thermal energy to the rest mass energy, relativistic effects become important. After comparing earlier non-relativistic calculations we extend the approach of Kruskal and Bernstein to take account of relativistic effects and also to investigate the influence of impurities. It is found that below the critical electric field ER = ED (kT/mec2) absolutely no runaways are generated. In addition, the number of runaway electrons produced by electric fields in excess of ER is calculated and we find significant modifications to the non-relativistic estimates when (ED/E)2 (kT/mec2) > 1.

A systematic procedure for studying the influence of kinetic effects on the stability of MHD ballooning modes is presented. The ballooning mode formalism, which is particularly effective for analysing high-mode-number perturbations of a plasma in toroidal systems, is used to solve the Vlasov-Maxwell equations for modes with perpendicular wavelengths on the scale of the ion gyroradius. The local stability on each flux surface is determined by the solution of three coupled integro-differential equations which include effects due to finite gyroradius, trapped particles, and wave-particle resonances. More tractable forms of these equations are then obtained in the low (ω < ωbi, ωti) and intermediate- (ωbi, ωti < ω < ωbe, ωte) frequency regimes with ωbj and ωtj being the average bounce and transit frequencies of each species. After further simplifying approximations, the kinetic results here are shown to be reducible to the MHD-ballooning-mode equations in the analogous limits, ω ≶ ωs where ωs = cs/Lc, with cs being the acoustic speed and Lc the connection length.

The sensitivity of the stability of the ideal n = 1 internal kink mode is analysed both analytically and numerically in rotating tokamak plasmas. These stability analyses have been carried out including the centrifugal effects of toroidal plasma rotation upon the equilibrium, and also inconsistently when the equilibrium is treated as static. The plasma stability is partially (consistent equilibrium) or wholly (inconsistent treatment) determined by the radial profiles of the plasma density and rotation velocity. It is found that the internal kink mode stability is strongly influenced by small variations in these plasma profiles. Indeed, modest perturbations to the profiles inside the q = 1 surface of only a few percent can result in a stabilising effect upon the kink mode with respect to the static mode growth rate becoming a destabilising effect at the same rotation amplitude, or vice versa. The implications of this extreme sensitivity are discussed, with particular reference to experimental data from MAST.

Recent gyrokinetic stability calculations have revealed that the spherical tokamak is susceptible to tearing parity instabilities with length scales of a few ion Larmor radii perpendicular to the magnetic field lines. Here we investigate this 'micro-tearing' mode in greater detail to uncover its key characteristics, and compare it with existing theoretical models of the phenomenon. This has been accomplished using a full numerical solution of the linear gyrokinetic-Maxwell equations. Importantly, the instability is found to be driven by the free energy in the electron temperature gradient as described in the literature. However, our calculations suggest it is not substantially affected by either of the destabilising mechanisms proposed in previous theoretical models. Instead the instability is destabilised by interactions with magnetic drifts, and the electrostatic potential. Further calculations reveal that the mode is not significantly destabilised by the flux surface shaping or the large trapped particle fraction present in the spherical tokamak. Its prevalence in spherical tokamak plasmas is primarily due to the higher value of plasma β, and the enhanced magnetic drifts due to the smaller radius of curvature.

Abstract. This paper reviews transport and confinement in spherical tokamaks (STs) and our current physics understanding that is partly based on gyrokinetic simulations. We show that equilibrium flow shear can sometimes entirely suppress ion scale turbulence in today's STs. Advanced nonlinear simulations of electron temperature gradient (ETG) driven turbulence, including kinetic ion physics, collisions and equilibrium flow shear, support the model that ETG turbulence can explain electron heat transport in many ST discharges.