10.7 Qualitative description of free-boundary instabilities 323whereλis spatially uniform. This equation is called the force-free equilibrium and its
solutions are helical vector fields, namely fields where the curl of the field is parallel to
the field itself. If a field is confined to a plane, then its curl will be normal to the plane
and so a field confined to a plane cannot be a solution to Eq.(10.140). The field must be
three-dimensional.
To summarize, it has been shown that current-driven instabilities arehelical and drive
the plasma towards a force-free equilibrium as prescribed by Eq.(10.140). Current-driven
instabilities are energized by the gradient ofJ 0 ‖/B 0 and become stabilized whenJ 0 ‖/B 0
becomes spatially uniform. Gradients inJ 0 ‖/B 0 can therefore be considered as the free
energy for driving kink modes. When all the free energy is consumed, the kinks are stabi-
lized and the plasma assumes a force-free equilibrium with spatially uniformJ 0 ‖/B 0.
This tendency to coil up or kink is a means by which the plasma increases its inductance.
However, when the plasma coils up into a state satisfying Eq.(10.140) it is in a stable
equilibrium. This stable equilibrium represents a local minimum in potential energy. There
might be several such local minima, each of which has a different value assketched in
Fig.9.2. As mentioned earlier this set of discrete energy levels is somewhat analogous
to the ground and excited states of a quantum system. Here, the vacuum magnetic field
is analogous to the ground state, while the various force-free equilibria (i.e., solutions of
Eq.(10.140)) are analogous to higher energy states.
10.7 Qualitative description of free-boundary instabilities
The previous section considered internal instabilities of a magnetofluid bounded by a rigid
wall so that no vacuum region existed between the magnetofluid and the wall. Let us
now consider the other extreme, namely a situation where not only does a vacuum region
exist external to the magnetofluid, but in addition, the location of the vacuum-magnetofluid
interface is not fixed and can move around. To focus attention on the effect of surface
motions, the simplest non-trivial configuration will be considered, namely a configuration
where the interior pressure is both uniform and finite. This corresponds to having∇P=
0 in the interior so that the entire pressure gradient and therefore the entireJ×Bforce is
concentrated in an infinitesimally thin surface layer.
The MHD energy principle showed that compressibility, manifested by having finite
∇·ξ, is stabilizing. Therefore, if a given system is stable with respect to incompressible
modes, it will be even more stable with respect to compressible modes, or equivalently, with
respect to modes having finite∇·ξ. Thus by assuming∇·ξ=0, the worst-case scenario is
considered and, in addition, the analysis is simplified. Finally, it isassumed that the plasma
is cylindrical and uniform in thezdirection (axially uniform). For example, the Bennett
pinch would satisfy these assumptions if all the current were concentrated on the plasma
surface. The physical basis of the two main types of current-driven instability, sausage and
kink, will first be discussed before engaging in a detailed mathematical analysis.
10.7.1Qualitative examination of the sausage instability
Consider a Bennett pinch (z-pinch), that is an axially uniform cylindrical plasma with an
axial current and an associated azimuthal magnetic field. In addition, asdiscussed above