10 Stability of static MHD equilibria
Solutions to Eq.(9.49), the Grad-Shafranov equation, (or to some more complicated coun-
terpart in the case of non-axisymmetric geometry) provide a static MHD equilibrium. The
question now arises whether the equilibrium is stable. This issue was forced upon early
magnetic fusion researchers who found that plasma which was expected to be well-confined
in a static MHD equilibrium configuration would instead became violently unstable and
crash destructively into the wall in a few microseconds.
The difference between stable and unstable equilibria is shown schematically in Fig.10.1.
Here a ball, representing the plasma, is located at either the bottom of a valley or the top of
a hill. If the ball is at the bottom of a valley, i.e., a minimum in the potential energy, then a
slight lateral displacement results in a restoring force which pushes the ball back. The ball
then overshoots and oscillates about the minimum with a constant amplitude because en-
ergy is conserved. On the other hand, if the ball is initially located at the top of a hill, then
a slight lateral displacement results in a force that pushes the ball further to the side so that
there is an increase in the velocity. The perturbed force is not restoring, but rather the op-
posite. The velocity is always in the direction of the original displacement;i.e., there is no
oscillation in velocity.
stable
equilibrium
unstable
equilibrium
Figure 10.1: Stable and unstable equilibria