Fundamentals of Plasma Physics

306 Chapter 10. Stability of static MHD equilibria

Jzand∂Bz/∂y∼Jx;this shear inhibits interchange instabilities. However, as will be

shown later this current can be the source for another class of instability called a current-

driven instability.

goodcurvature good curvature

bad curvature

mirrorfield cuspfield

Figure 10.4: Mirror fields have bad curvature at the mirror minimum, good curvature near
the maximum;cusp fields have good curvature everywhere.

10.3 The MHD energy principle

Suppose a system initially in equilibrium is subject to a small perturbationinstigated by

random thermal noise. The perturbation will affect all the various dependent variables in

a way which must be consistent with all the relevant equations and boundary conditions.

Thus, the perturbation may be considered as a low-level excitation of someallowed normal

mode of the system. The mode will be unstable if it reduces the potential energy ofthe sys-

tem and, if unstable, the mode growth rate will be proportional to the amount by which the

potential energy is reduced. In order to investigate whether a given system is stable, it suf-

fices to show that no modes exist which reduce the system potential energy. Demonstrating

MHD stability this way was first done by Bernstein, Frieman, Kruskal andKuslrud (1958)

and the method is called the MHD energy principle.

Each mode has its own specific pattern for displacing the magnetofluid volume elements

from their equilibrium positions. The pattern of displacements can be represented by a

vector function of positionξ(x)that prescribes how afluid volume element originally

at locationxis displaced. Thus, the mode involves moving afluid element initially at

xto the positionx+ξ(x). The displacement of a magnetofluid element initially at the

surface is shown in Fig. 10.5, a sketch of a two-dimensional cut of a three dimensional

magnetofluid (plasma) surrounded by a vacuum region in turn bounded by a conducting

wall. The wall could be brought right up to the magnetofluid to eliminate the vacuum

region or, alternatively, the wall could be placed at infinity to represent a system having no

wall and surrounded by vacuum.