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*Astron. Astrophys. 355, 818-828 (2000)*
## 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.
Resonant surfaces are crucial for the understanding of the CDI.
They are magnetic surfaces where the condition
holds, or equivalently,
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
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