One set of experiments provides an overview of the kind of physics issues that we want to study. In these experiments, shears in the initial rotational flow drive Kelvin-Helmholtz (or diocotron) instabilities, generating vortical structures, which are then destroyed by vortex merger and by shears in the background rotational flow. During this process, the plasma undergoes rapid transport to a 2D meta-equilibrium that is stable, quiescent, and long lived. This meta-equilibrium lasts for about a second, after which 3D processes due to field errors or viscosity cause it to evolve further.
The challenge to theory is to predict the density distribution for this meta-equilibrium state. Our first approach to this problem was based on the hypothesis that the plasma evolves to a state of maximum 2D entropy S=-INT(d²r n*lnn), subject to fixed values for the particle number, canonical angular momentum, and electrostatic energy. These quantities are conserved during the evolution. This hypothesis is based on a picture where the turbulence is sufficiently violent to shred the density distributions into filaments. Small pieces of each filament are assumed to disassociate and to wander ergodically over the region of the allowed phase space. We represent each of these small fluid elements by a point vortex, and calculate the mean-field entropy subject to fixed values of the three conserved quantities. We have carried out a similar analysis using small but finite size vortices that cannot overlap. This takes into account the incompressible nature of the flow.
Another variational approach to this problem is based on the hypothesis that the enstrophy Z_2==INT(d²r*n²) approaches a minimum value subject to fixed values of the particle number, canonical angular momentum, and energy. This "selective decay" hypothesis has been used traditionally in discussions of 2D turbulence. Formally, the 2D drift-Poisson equations conserve the enstrophy, as well as all other moments Z_n. However, as the turbulence produces ever finer spatial variation in n, any course-grain average or viscous effect makes the enstrophy decrease.
Experimentally, we find that minimization of enstrophy best describes the meta-equilibria
arising from instability-driven turbulence on initially hollow electron columns. For low
energies, both the measured meta-equilibrium density (vorticity) profiles n(r) and the
measured enstrophy are in close agreement with predictions based solely on the measured
number of particles (circulation), angular momentum P_
, and energy H
.
Figure 6 shows the measured final Z_2 versus the measured excess energy
H^(exc)
H-H^(min), and the prediction of theory(curve).
At higher energies, H^(exc)
1.6*10¯³, the theory breaks down, but can be
extended if non-monotonic or non-axisymmetric states are allowed. We intend to
characterize these states for comparison to experiments.
Future theory and experiments will ascertain the limits of applicability of the enstrophy
minimization model. We are now able to accurately image higher excess energy (i.e. more highly
filamented) flows, and we observe a number of interesting classes of meta-equilibria. As seen in
Fig.7, a finely filamented initial condition forms dozens of small vortices within about one
plasma rotation time 
_R. Like-signed vortices then merge and shed new filaments, cascading
the energy to long wavelengths, and the entropy (and enstrophy) to fine scales where dissipation
necessarily occurs. Finally, the turbulence self-organizes to a sheared, stable meta-equilibrium,
with some level of decaying "noise" from holes or clumps (600 _R). For many initial
conditions, we observe that the longest-lived vortices are self-trapped depressions in the
electron density, i.e., "holes." This stability of holes and clumps in shear flows is thought
to explain persistent features such as the Great Red Spot of Jupiter, but few experiments
are able to check the theories. The last frame of Fig.7 shows a meta-equilibrium which is highly
peaked (4:1) in the center, with two long-lived holes.
Recently, long-lived "vortex crystal" states have been observed for similar highly filamented initial conditions with low internal shear, as shown in Fig.8. Here, the overall relaxation is arrested when individual vortices settle into a stable, rotating crystalline pattern which is presumably a local lowest energy configuration. The process by which the individual vortices lose energy to the diffuse background of vorticity is presently unknown but experimentally quantifiable. Theoretical studies of the energy loss mechanism and of the crystal structure will be pursued through 2D numerical simulations. Experimentally the individual vortices decay in place on the time scale of about 1000 bulk turnover times, decreasing in number and re-arranging in position. The decay of individual vortices presumably represents some form of plasma diffusion or viscosity, and we hope to fully characterize it experimentally and theoretically.
This evolution data is fully quantitative, and precise comparisons to theory can be made.
The measured z-integrated density n(r,,t) is routinely converted to the full
3D n(x,y,z,t) by solving the self-consistent Poisson's equation for (r,,z,t),
subject to the known boundary conditions. Integrals such as energy, angular momentum, circulation,
enstrophy, and entropy are then straightforward sums over the data. We believe that this tightly
coupled program of experiments and theory will give significant insights into self-organization of
2D turbulence.