Fundamentals of Plasma Physics

326 Chapter 10. Stability of static MHD equilibria

azimuthal field is weaker so that the convex bulge will tend to increase.This is just the

hoop force trying to increase the major radius of a curved current. If the externally imposed

axial magnetic field is sufficiently strong, the energy required to bend the external field will

exceed the free energy available from the kinking and the instability will be prevented.

10.7.3Qualitative examination of wall stabilization

Now consider a plasma with no externally producedBzfield but surrounded by a close-

fitting perfectly conducting wall. The plasma axial current generatesan azimuthal field in

the vacuum region between the plasma and the wall. If an instability causes the plasma to

move towards the wall, then the vacuum region becomes thinner and the magneticfield in

the vacuum region becomes compressed since it cannot penetrate the wall or the plasma.

The increased magnetic pressure acts a restoring force, pushing the plasma back away

from the wall. Thus, a close-fitting perfectly conducting wall can stabilize both kinks and

sausages.

surface

current

perfectly conducting

wallatr
b

a−a
 b

uniform

pressure

plasma

vacuum

uniform

pressure

plasma

region

vacuum

region

Pressure

surface

current

region

r

Figure 10.8: Equilibrium for free-boundary stability analysis: assumed cross-section (left)
and assumed pressure profile (right).

10.8 Analysis of free-boundary instabilities

Consider the azimuthally symmetric, axially uniform cylindrical plasma of radiusashown

in Fig.10.8. The plasma is subdivided into two concentric regions consisting of(i) the in-

terior region 0 <r<a−and (ii) a thin surface layer a−<r<a+. In the limit of

infinitesimal surface thickness,a−approachesa+.The interior pressureP 0 is uniform and

the interior current density is zero so that bothJ×Band∇Pare zero in the interior. Thus,

a finite current density existsonlyin the surface layer and everywhere else the magnetic

field is a vacuum magnetic field. The plasma is also assumed to be surrounded by a per-

fectly conducting wall located at a radiusbwherebis larger thana.The MHD equilibrium