[en] The formalism for internal ballooning modes in a tokamak is extended to retain the strong radial variation of the ion diamagnetic drift frequency characteristic of edge plasmas in the pedestal region. The resulting finite Larmor radius (FLR) stabilization is modified and can be weaker than in the case of constant diamagnetic frequency

[en] A number of experiments carried out in several Reversed Field Pinches (see, e.g. and references therein) have shown that plasma in such devices can settle into so-called Quasi-Single-Helicity (QSH) states. These states are characterized by a magnetic spectrum in which the amplitude of the mode with poloidal wave number m = 1 and with the innermost resonant surface is much larger than the amplitude of all the other modes. QSH states can often exhibit a cyclic behavior in which, at each cycle, the amplitude of the dominant helical mode initially grows in time, then saturates and eventually abruptly decays. QSH states are of interest not only because they represent an intriguing self-organization process but also because their occurrence can correspond to an improvement in the particle confinement properties of the device. However, a complete theoretical explanation for the occurrence of QSH states is still missing. In this contribution we propose that QSH states might emerge as consequence of a small deviation of the mean magnetic field from the linear force-free state predicted by the classical theory by Taylor. In particular we show that, in cylindrical geometry, force-free equilibria for which the parameter μ, defined as the ratio between the parallel current density and the magnetic field, is a step function of the radius, can be linearly tearing unstable with respect to the innermost resonant mode with m = 1 but stable with respect to all the other modes. A scenario is then depicted according to which a plasma initially relaxed to a tearing-stable Taylor state could subsequently become tearing-unstable and lead to a QSH state if even a small step in μ forms, due for instance to a peaking of the current density caused by core ohmic heating. The results referring to the linear phase are complemented by nonlinear results related to the saturation phase. We derive an analytical relation that predicts the saturated island width ω as function of the parameters of the initial stepped-μ equilibrium. In particular it is found that, for small steps, ω grows almost linearly with the step height Δμ but the dependence becomes nonlinear as Δμ increases. (Author)

[en] An axially symmetric plasma immersed in a poloidal magnetic field with closed lines is considered. Low-frequency electrostatic modes are studied kinetically for an ''intermediate collisionality'' ordering, in which the particle collision frequency is much smaller than the transit or bounce frequency, but much larger than the mode, magnetic drift, and diamagnetic drift frequencies. This ordering is appropriate for the Levitated Dipole Experiment (LDX) [J. Kesner , 17th IAEA Fusion Energy Conference, Yokahama, Japan (IAEA, Vienna, 1999)] and some other closed field line devices. ''High-frequency'' magnetohydrodynamic-like and ''low-frequency'' entropy modes are found and stability boundaries are determined. Collisional effects are considered and the corresponding ion gyro-relaxation effects are evaluated. These effects introduce dissipation (or inverse dissipation) and are shown to modify the stability picture considerably, while leaving large stability regions in the d, η parametric space, where η is the ratio of the gradients of temperature and density and d is the ratio of the diamagnetic and magnetic drift frequencies

[en] The effect of the inductive electric field of a tokamak on the parallel (and poloidal) ion flow in the banana regime is evaluated. It is demonstrated that the flow is in the direction of the parallel current and is surprisingly large -- comparable to the usual banana regime ion temperature gradient drive

[en] The saturation of the tearing mode in a plasma column is investigated in the framework of the resistive magnetohydrodynamics approximation. In particular, a perturbative procedure is adopted to evaluate the structure of the magnetic island in three relevant physical conditions, depending on the model for the evolution of the resistivity, which may be affected by the growth of the mode. In cylindrical geometry, which is well suited to describe a large-aspect-ratio, low-beta tokamak plasma, the magnetic island is asymmetric with respect to the magnetic surface where reconnection occurs. New relations for the saturated island width w_s as a function of the relevant features of the equilibrium current density profile, i.e., its gradient and curvature at the reconnecting surface, are obtained. Finally, equivalent relations are also derived in the slab limit

[en] 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 stabilizing ion orbit effect, obtained previously by Cowley et al (1986 Phys. Fluids 29 3230), which implies large values of the tearing mode stability parameter Δ′ 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 stabilization for all Δ′ due to the shielding effects of the electron temperature gradients, extending the result of Drake et al (1983 Phys. Fluids 26 2509) 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 (1989 Phys. Fluids B 1 364) 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 Δ′ 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. (paper)

[en] This paper describes work on an important class of plasma instabilities in toroidal confinement systems. These are instabilities with large mode number in the toroidal direction but long wavelength parallel to the magnetic field. After a brief historical introduction the now standard method-the ballooning representation-for calculating such modes is described. This method is remarkably successful for most stationary plasmas, but breaks down for configurations with low magnetic shear or with significant sheared rotational flow. These are two areas of great current interest because of their association with 'transport barriers' in tokamaks. Some extensions of ballooning theory for dealing with these situations are described and some preliminary results, in particular showing the stabilizing effect of sheared rotation, are presented

[en] Many models for anomalous transport consider the turbulent E × B transport arising from electrostatic micro-instabilities. In this paper we investigate whether the perturbed magnetic field that is associated with such instabilities at small but finite values of β can lead to significant stochastic magnetic field transport. Using the tearing parity, long wavelength ion temperature gradient (ITG) modes in a plasma slab with magnetic shear as an example, we calculate the amplitude of the perturbed magnetic field at the resonant surface that results. The plasma model consists of a Braginskii description for the electrons with the dissipation at the resonant surface evaluated for the semi-collisional regime, while a finite ion Larmor radius kinetic model is invoked for the ions. The resulting stochastic field transport is estimated and also compared with an estimate for the E × B transport due to the ITG mode. (paper)

[en] The stability of drift waves in configurations with low magnetic shear, s, including the case with a minimum in the safety factor, q_min, which is a situation relevant to the formation of internal transport barriers, is analysed. First the small s limit of the ballooning representation is shown to, indeed, coincide with the result of a shear free calculation. Then, the limitations associated with using the ballooning transformation at low shear are discussed. Radially extended ballooning modes can be considered to result from the toroidal coupling of 'modelets' (each of which can contain a number of poloidal harmonics) centred on adjacent resonant surfaces. In a region of low magnetic shear, a modelet situated on a particular resonant surface does not extend to the adjacent surface and therefore the coupling of these is weak and the ballooning theory fails. An alternative approach based on a recurrence relation between modelets is used to analyse these situations with small s. This is then applied to electron drift waves and ion temperature gradient modes as examples. These studies show that for sufficiently long wavelengths, radially extended ballooning modes cannot exist close to a point of zero magnetic shear; however, independent modelets can still exist at resonant surfaces near q_min. The criteria on the wavelength for this situation to prevail are presented. The example of electron drift waves centred on q_min is analysed and it is found that, even for this 'slab-like' electron drift wave, damping from an outgoing wave can be entirely suppressed

[en] We propose a model for edge localized mode (ELM) evolution which goes beyond linear stability arguments by hypothesizing that peeling modes initiate a Taylor relaxation (a constrained minimization of the magnetic energy) of an outer annular plasma region. The relaxation has two effects on peeling mode stability: (a) As the relaxation process proceeds radially inwards it leaves in its wake a Taylor state, which for conventional tokamak ordering is simply a flattened equilibrium toroidal current density. This effect acting in isolation would provide a destabilizing effect (for conventional current profiles the edge current density would increase); (b) The formation of a (negative for conventional current profiles) skin current at the plasma-vacuum interface which has a counteracting stabilizing effect on peeling modes. For a finite relaxed annulus, these two opposing effects can balance and give a configuration that is stable to all possible peeling instabilities. The radial extent of the relaxed region required for stability can be calculated using this balance. This then leads to model predictions for ELM characteristics such as the widths and mode numbers, the magnitude of the attendant energy losses and the natural (deterministic) scatter in these quantities. We compare these model predictions with a number of experimentally observed ELM properties. Further, expanding the governing equations gives analytic expressions for ELM widths in terms of localized edge parameters. Peeling modes can occur even when the critical pressure gradient for the onset of ballooning modes has not been reached. For this reason 'type III' ELMs, which typically occur just above the threshold for L-H transitions, may be best described by this model