Pure electron plasmas in Penning traps have excellent confinement properties. Also, certain vortical structures in neutral fluids are particularly robust and long-lived. We want to determine the necessary condition for this longevity, or good confinement. Our goal is to extend good confinement to new confinement geometries (e.g., toroidal traps), and even to neutral or partially neutral plasmas (e.g., electron-positron plasmas).
The first step in this program is the development of a new stability theorem. A preliminary
version of this theorem has been published, but the theorem continues to evolve in generality.
In our current thinking, the theorem applies both to traps with open field lines (e.g., Penning
traps) and to traps with closed field lines (e.g., toroidal traps). It is assumed that the
plasma dynamics can be described by bounce-average (or toroidal-average) guiding center drift
dynamics. This requires that the dynamical frequencies be ordered as 
_c>>
_b>>_r,
where _c is the cyclotron frequency, _b is the axial-bounce (or toroidal rotation)
frequency, and _r is the drift-rotation frequency. In this case the plasma kinetic energy
is bound up in the adiabatic invariants µ=m*V_perp²/2B and I=INT(dl*V_parr). Also,
the flux linked by an element of plasma is conserved during the evolution, that is, the cross-field
flow is a 2D incompressible flow when referred to flux coordinates.
Thus, a plasma equilibrium is stable against low-frequency perturbations that respect the drift dynamics (e.g., diocotron modes) if the state has maximum energy as compared to neighboring states that are accessible under incompressible flow. Of course, the plasma cannot change out of the state of maximum energy because energy is conserved during the evolution and there are no locally accessible states with the same energy. It is possible to find states of maximum energy because the kinetic energy is bound up in the adiabatic invariants and cannot increase arbitrarily. The great advantage of the new theorem, relative to previous thinking about the stability of non-neutral plasmas is that cylindrical symmetry of the trap and equilibrium is not required.
Recent experiments have provided support for the new theorem. These experiments were carried out in Penning traps with uniform magnetic field. In the first experiment, we produced an asymmetric state simply by displacing the column off-axis, that is, by exciting an m=1 diocotron mode. We had proven that such a state is maximum energy, and experimentally the state was observed to be stable and very long-lived (10^5 periods~second). In a set of experiments at Berkeley, stationary asymmetric equilibria were produced by applying asymmetric potentials on the wall. These equilibria were observed to survive stably for long periods (second) and the observed cross-sectional shapes were predicted by maximizing the electrostatic energy.
Encouraged by the experiments, we decided to consider new confinement geometries. As a first example, we have been considering toroidal confinement geometry. For a non-neutral plasma, the confining magnetic field can be purely toroidal, since the poloidal ExB drift rotation effectively provides the rotational transform. Because the toroidal curvature spoils any cylindrical symmetry about the direction of the magnetic field, the new theorem is needed. For the simple case of a water bag model, Fig. 9 shows the plasma boundary for several maximum energy states.
It is interesting to note that positrons would respect the adiabatic invariants just as well as electrons, and it is natural to ask if a partially neutral electron-positron plasma could be stable in the toroidal trap. When only ExB drifts are retained and the electrons and positrons are uniformly mixed, the theorem says yes, since the two species simply evolve as a single non-neutral fluid. When other drifts are included, the answer is not so clear. For example, the centrifugal force associated with the poloidal rotation can drive a flute instability. On the other hand, shear in the rotational flow may smooth out the charge separation associated with the flute and stabilize it. We intend to explore these issues both analytically and numerically.
It is clear from our experiments with two or more stable vortices that the stability theorem is
not as general as it should be: some perfectly stable states are not maximum energy states. We
believe that the dynamics is constrained by an additional adiabatic invariant that is associated
with the rapid rotation of a vortex about its own axis. We plan to explore this issue theoretically.
Also, we observe that plasmas with _b>>_r (rigid 2D plasmas) have better confinement properties
than plasmas with _b
_r (floppy plasmas). On floppy plasmas, asymmetric applied fields have
been observed to excite a decay instability with a daughter plasma wave. From the perspective
of the stability theorem, increasing _r to the range of _b brings the parallel degrees
of freedom into resonance with the rotational motion and allows energy to be deposited in these
degrees of freedom, specifically in a plasma wave. We plan to explore the distinction between
floppy and rigid plasmas in light of the stability theorem.