10.5 Current-driven instabilities and helicity 31910.4 Discussion of the energy principle
The terms in the integrands ofδWF,δWint,andδWvacare of two types: those that are
positive-definite and those that are not. Positive-definite terms always increaseδWand
are therefore stabilizing whereas terms that could be negative are potentially destabilizing.
Consideration ofδWFin particular shows that magnetic perturbations interior to the mag-
netofluid and perpendicular to the equilibrium field are always stabilizing, since these per-
turbations appear in the formB^21 ⊥. It is also seen that a positive-definite term∼(∇·ξ)^2
exists indicating that a compressible magnetofluid is always more stable than an otherwise
identical incompressible magnetofluid. Hence, incompressible instabilities are more vio-
lent than compressible ones. It is also seen that two types of destabilizing terms exist. One
gives instability if
(κ·ξ⊥)(ξ⊥·∇P 0 )>0; (10.117)this is a generalization of the good curvature/bad curvature result obtained in the ear-
lier Rayleigh-Taylor analysis. In the vicinity of the magnetofluid surface all quantities
in Eq.(10.117) point in the same direction and so Eq.(10.117) can be written as
κ·∇P 0 >0=⇒instability (10.118)which gives instability for bad curvature (κparallel to∇P 0 )and stability for good curva-
ture (κantiparallel to∇P 0 ). A Bennett pinch has bad curvature and is therefore grossly
unstable to Rayleigh-Taylor interchange modes. Instabilities associated with Eq.(10.117)
are called pressure-driven instabilities, and are important in plasmas where there is signifi-
cant energy stored in the pressure (highβwhereβ=2μ 0 P 0 /B 02 ).
The other type of destabilizing term depends on the existence of a force-freecurrent
(i.e.,J 0 ‖ =0)and leads to internal kink instabilities. These sorts of instabilitiesare called
current-driven instabilities (although strictly speaking, only the parallel component of the
current is involved).
There also exist instabilities associated with the magnetofluid-vacuum interface. The
energyδWintcan be thought of as the change in potential energy of a stretched membrane
at the magnetofluid-vacuum interface where the stretching force is given by the difference
between the vacuum and magnetofluid forces pushing on the membrane surface;instability
will occur ifδWint< 0. When investigating these surface instabilities it is convenient to
setδWFto zero by idealizing the plasma to being incompressible, having uniform internal
pressure, and no internal currents. Surface instabilities can exist only if the surface can
move, and so require a vacuum region between the wall and the plasma. Thus, moving a
conducting wall right up to the surface of a conducting plasma prevents surface instabilities.
These surface instabilities will be investigated in detail laterin this chapter.
10.5 Current-driven instabilities and helicity
We shall now discuss current driven instabilities and show that these are helical in nature
and are driven by gradients inJ 0 ///B 0 .To simplify the analysisP→ 0 is assumed so that
pressure-driven instabilities can be neglected since they have already been discussed. On
making this simplification, Eq.(10.97) shows that the parallel component of the perturbed