Previous work by Johnson and Greene on resistive instabilities is extended to finite‐pressure configurations. The Mercier criterion for the stability of the ideal magnetohydrodynamic interchange mode is rederived, the generalization of the earlier stability criterion for the resistive interchange mode is obtained, and a relation between the two is noted. Conditions for tearing mode instability are recovered with the growth rate scaling with the resistivity in a more complicated manner than η3/5. Nyquist techniques are used to show that favorable average curvature can convert the tearing mode into an overstable mode and can often stabilize it.

Resistive instabilities in general toroidal plasma configurations

The transition of the tearing instability from the collisional to the collisionless regime is investigated kinetically using a Fokker–Planck collision operator to represent electron‐ion collisions. As a function of the collisionality of the plasma, the tearing instability falls into three regions, which are referred to as collisionless, semi‐collisional, and collisional. The width Δ of the singular layer around k⋅B0=0 is limited by electron thermal motion along B0 in the collisional and semi‐collisional regimes and is typically smaller than ρi, the ion Larmor radius. Previously accepted theories, which are based on the assumption Δ≫ρi, are found to be valid only in the collisional regime. The effects of density and temperature gradients on the instabilities are also studied. The tearing instability is only driven by the temperature gradient in the collisional and semi‐collisional regimes. Numerical calculations indicate that the semi‐collisional tearing instability is particularly relevant to present day high temperature tokamak discharges.

Kinetic theory of tearing instabilities

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.

https://iopscience.iop.org/article/10.1088/0029-5515/18/1/010/pdf

Hydromagnetic stability of tokamaks

A quasi‐linear analytical model is used to describe the nonlinear growth and saturation of tearing modes with mode number m⩾2. The saturation of the magnetic island growth is the quasi‐linear development of a single mode rather than a mode coupling process. The saturation amplitude, which is dependent on the form of the resistivity, is in good agreement with results obtained previously by numerically advancing the full set of nonlinear equations.

Saturation of the tearing mode

A principal pressure limit in tokamaks is set by the onset of neoclassical tearing modes (NTMs), which are destabilized and maintained by helical perturbations to the pressure-gradient driven “bootstrap” current. The resulting magnetic islands break up the magnetic surfaces that confine the plasma. The NTM is linearly stable but nonlinearly unstable, and generally requires a “seed” to destabilize a metastable state. In the past decade, NTM physics has been studied and its effects identified as performance degrading in many tokamaks. The validation of NTM physics, suppressing the NTMs, and/or avoiding them altogether are areas of active study and considerable progress. Recent joint experiments give new insight into the underlying physics, seeding, and threshold scaling of NTMs. The physics scales toward increased NTM susceptibility in ITER, underlying the importance of both further study and development of control strategies. These strategies include regulation of “sawteeth” to reduce seeding, using static...

http://absimage.aps.org/image/DPP05/MWS_DPP05-2005-000592.pdf

Neoclassical Tearing Modes and Their Control

Detailed computational results are presented on the stability and radial distribution of linear tearing modes in cylindrical tokamaks of various radial profiles. In the case of a skin‐current profile, a “double tearing mode”, with two points of discontinuity in the radial magnetic field perturbation is found. An analytical method is also derived for comparison of the stability of different radial profiles. It is further shown that the tearing mode can be driven by finite electron viscosity, as well as by the usual finite resistivity mechanism.

H. Selberg

Papers

Tearing mode in the cylindrical tokamak