4. Current-driven instabilities of the linear pinch
A straight cylinder of length with periodic boundary conditions serves also as a model to study the stability of toroidal configurations with large aspect ratio, , designed for thermonuclear fusion, such as the TOKAMAK or the RFP. We have argued in the previous section that CDI are absolute instabilities in the restframe of the magnetized jet, and we therefore very briefly review some of the results obtained for the stability of the static linear pinch that are relevant for our purpose. The most dangerous instability is the current-driven kink instability, due to field-aligned currents, while the axisymmetric pinch modes are stabilized by the absence of a negative pressure gradient.
inside the plasma column, with the wavevector of the instability and the equilibrium magnetic field. A configuration with resonant surfaces somewhere in the plasma is unstable w.r.t. internal modes, and we will refer to these instabilities also as resonant modes. Consequently, the central core of an infinitely long configuration with radially increasing pitch (where ) is always unstable for sufficiently long wavelengths, and the modes are generally resonant for an interval in k with ( being the limiting wavenumber). A TOKAMAK is also characterized by an increasing q-profile (), but the toroidal topology only admits discrete wavenumbers . Consequently, implies stability w.r.t. modes (Kadomtsev 1975). Configurations with an increasing pitch profile and central values will in the following sometimes be referred as TOKAMAK-like. The most unstable mode in this case is generally resonant.
The perturbed field is related to the fluid displacement through Faraday's law, , which expanded in normal modes yields for the radial component , i.e. at the resonant surface the radial component of the perturbed magnetic field vanishes. In the case of an increasing pitch profile (and for given k and m) the eigenfunctions can, at marginal stability, be approximated by a step function, which is constant within and vanishes outside the resonant surface. The resulting deformation due to a mode is then a helical distortion of the jet core inside the resonant radius. With increasing growth rate the eigenfunctions become smoother due to inertial effects. The resonances are particularly important for the understanding of the the nonlinear regime of the internal kink mode of the TOKAMAK (Rosenbluth et al. 1973), where a singular current sheet is forming at the resonant surface. In the presence of a small non-vanishing resistivity, the current sheet diffuses and drives a magnetic reconnection process.
Not all current-driven instabilities are resonant, however. In particular, a class of fusion devices called reversed field pinches (RFP; Bodin & Newton 1980), is characterized by a decreasing pitch profile, which can slightly reverse its sign near the edge. The most unstable (i.e. with the fastest growth) ideal mode (again ), is non-resonant from above (Caramana et al. 1983) since everywhere. Unlike the TOKAMAK, no practical necessary and sufficient criteria are available for this type of configuration, and in particular no critical wavenumber where the instability sets in can be given a priori. These instabilities are generally characterized by a somewhat broader and smoother eigenfunction than the resonant modes, and they do not vanish at the conducting wall. More important, everywhere, and they do not develop singular current sheets in the nonlinear regime (Caramana et al. 1983). For equilibria with monotonically decreasing pitch ( in Eq. (5)) the most unstable mode is non-resonant, similar to the RFP. But note that in spite of its decreasing pitch variation, the dominant kink mode of the unstable BFM configuration can also be a resonant ( in this case) instability, e.g. in the case . The resonant surface, however, is localized in a region of negative pitch.
Finally, magnetic shear, , is favourable for the stability of a pinch. Shear and pressure gradient vanish on the axis. This leads to a local stability criterion involving the second derivative of the pitch on the axis (cf. Bodin & Newton 1980),
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000