When azimuthally-asymmetric potentials are applied to the wall of a Penning-Malmberg trap, the plasma deform into a stationary cylinder of noncircular cross-section. Such highly deformed, stationary non-neutral plasma columns are unexpectedly long-lived. Normally, non-neutral plasmas are stored in Penning-Malmberg traps with cylindrically-symmetric wall boundary potentials, and the equilibrium plasma shape is a symmetric cylinder. Wall potentials can be applied by biasing isolated sections of the wall surrounding the plasma.

A typical deformed shape, found experimentally is shown below.

Since the wall potentials are no longer symmetric, angular momentum conservation is no longer guaranteed, and the standard justification for the long lifetime of non-neutral plasmas is no longer applicable. Consequently, the long lifetimes of these deformed plasmas was a surprise.

Theoretical study of these deformed plasmas begins with understanding
the equilibrium conditions. Chu *et. al.* studied the equilibrium
shapes of the slightly deformed plasmas that result from small wall potential
perturbations. The results of Chu’s analytic (left) and numeric (center)
models for the experimental plasma at the right are shown below:

Recently we have been studying highly deformed plasmas such as the one below. In principle, and shape even one as deformed as the one below, can be put in equilibrium

All deformed plasmas can be put in equilibria, but not all deformed
plasmas are in stable equilibria. When the deformation is small, the plasma
is in a *maximal* energy state. Maximal energy states (like a ball
at the top of a mountain) are normally thought to be unstable. But in our
system, potential energy cannot be turned in kinetic energy, so a maximal
energy state is actually stable. Hence the plasma are typically in stable
equilibria. But if the plasma is deformed to much, the stability will bifurcate.
For example, this
movie shows the effect of applying an ever greater cos(2q)
wall potentials to an experimental plasma. Click on the plasma to
get a more informative movie. Determining the stability of more complicated
shapes is a difficult problem in advanced bifurcation theory. The
symmetry of the system has a strong influence on the type of bifurcation,
and many high order bifurcations are predicted to occur.

More information on equilibria and stability can be found in the papers
in the bibliography: